text:
What's On In Derby
Things to do in Derby
Kids Sports Clubs
Zoo & Animal farms
Quad Bike Tours
Paintballing
Walk Tours
Guest Houses, B&B's
Pickford's House Museum
National Tramway Museum
Museum & Art Gallery
Derby Arboretum
Haddon Hall
Heights of Abraham
Dentist Services
Derby News & Events
What's on in Derby is the "go to" website packed with information for residents and visitors to make your time in Derby the best it can be. Great ideas on what to do and see, the top places to eat and stay, attractions, events, news and more…
The best Sports Facilities, Family and Kids activities in and around Derby
The best Shopping and Local Services in and around Derby
Home Derby News
Live updates as Government to announce Derbyshire's Covid-19 tier today
Restaurant owners have their say on impact of possible tier 3 extension
Restaurant owners in Derbyshire have spoken out about the potential impact of a possible extension to tier 3 restrictions.
Nawa, owner of the Himalayan Gurkha Restaurant, in Derby, claimed it would be unfair to keep dining venues closed while other areas of the economy remain relatively unaffected.
He said: "We are actually very much hoping to open the restaurant and would feel really disappointed if we can't, as all the non-essential shops, barbers and beauty shops are open whereas eating in has been left behind.
"It would feel even more disappointing as the town centres are packed with people where social distancing is hard to maintain.
"We are struggling as we are losing out so much and it's getting harder to remain open only for takeaways."
However, for Holly Faulkner, joint owner of Derby-based The Wonky Table, lifting restrictions now would still be a case of too little, too late.
Holly said it would be difficult to prepare for a move out of tier 3, and that her staff would have to work overtime to make reopening worthwhile.
She said: "We've not been taking bookings and we haven't put up Christmas decorations in the restaurant because we just can't predict what's going to happen.
"Our staff numbers are limited because we've had to keep some people on furlough, so some of us would have to pull 14-hour shifts to make it work.
"We'd also have to sort stuff out with our suppliers pretty quickly. It hasn't been worth the risk to order food supplies if we don't know we're going to need it.
"How can you prepare when you don't know what's going on?"
You can hear from more restaurant owners in our full story here: Restaurants have their say on impact of possible tier 3 extension
The Wonky Table Restaurant in Derby (Image: Google)
Courtesy of Derbyshire Live – Derby News
Topics:Derby News
Derby and Derbyshire are placed in tier two restrictions
Travellers appear to set up camp in Ascot Drive
Killer on the loose in Derbyshire as trio abscond from prison
Derby pub landlord working as delivery driver to save business
Derbyshire coronavirus deaths up by 29 as cases continue to climb
The Government said a further 1,280 people had died across the UK Courtesy of Derbyshire Live – Derby News…
Derby attempted murder accused told medics 'I did it, I strangled her'
Bradley Souto's 999 call was played at his trial Courtesy of Derbyshire Live – Derby News…
Live updates ahead of tougher lockdown policing
It comes as cases continue to rise Courtesy of Derbyshire Live – Derby News…
Showcase cinema in Derby has closed down for good
It will not reopen after shutting in September Courtesy of Derbyshire Live – Derby News…
Derbyshire Live readers overwhelmingly agree with return to national lockdown
According to our survey 88% of readers are in favour of the move Courtesy of Derbyshire Live – Derby News…
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NWT Pottery Barn Darcy Rug. The rug had been removed from the original packing, but has no noticeable flaw. The item "NWT Pottery Barn Darcy Rug 5'x8" is in sale since Thursday, March 28, 2019. This item is in the category "Home & Garden\Rugs & Carpets\Area Rugs". The seller is "jaxluv2010" and is located in Merrimac, Massachusetts.


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SERVICE=${APPLICATION_NAME}

if [ ${APPLICATION_NAME} = "themeBlueprint" ] || [ ${APPLICATION_NAME} = "theatreBlueprint" ]; then
	SERVICE="the-wall"
fi

file="/etc/init/${SERVICE}.conf"
if [ ! -f "$file" ]; then
  # fall back to the default
  SERVICE="primaryApplication"
fi

echo "Stopping $SERVICE (${APPLICATION_NAME})"
service $SERVICE stop

redpajama_set_name: RedPajamaGithub


You now have one more option when it comes to shopping for groceries in West Palm Beach.
The Aldi along Okeechobee Boulevard near Congress Avenue is back open.
After a six week re-modeling project the store held a grand re-opening event today. The store is giving away free produce for a year to one lucky winner. You can sign up in person until Sunday.
The West Palm Beach store is part of an $1.9 billion Aldi investment to remodel and expand more than 1,300 stores.


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The Best Bars to Check Out in 23 Brooklyn Neighborhoods
By Dillon Mafit Published On 10/17/2017
D. Cyrus / Flickr
If Manhattan disappeared into the sea, Brooklyn would be the third most populous city in the United States. At 71 square miles, its sheer size dwarfs the island of Manhattan, and it boasts nearly fifty distinct neighborhoods, from the traditionally Polish Greenpoint in north Brooklyn, to the once Norwegian shores of Bay Ridge. With this guide, you'll never be without a place to get a good stiff drink. Here, the best bars to go to in 23 different Brooklyn neighborhoods.
This raucous, ersatz metal bar embodies Williamsburg's hipster/grunge/forever-young ethos better than any other bar in the hood. They have a slammin' jukebox, the greatest Michelada in all of Brooklyn (served with a mini Miller High Life), and they serve their beers in giant goblets fit for a Norse god (or Lemmy Kilmister). Oh, and if you're a fan of the super-boozy coffee slushie thing at Skinny Dennis (same owners, same hood), they serve it here as well.
Uber cozy, you can spend hours lingering over a drink at this off-the-beaten-path Greenpoint bar. There are killer cocktails like The Countdown (tequila, sweet vermouth, yuzu), Boilermaker deals, a dense backbar, oysters, a fireplace (!) and intimate live shows (or DJs). It's everything you could ever want out of a neighborhood bar.
Keith Kenji's The Word, a variation on the Last Word made with jalapeno-infused tequila and Laphroaig scotch, is piquant, campfire-rific and refreshingly chuggable year round. After you drink two or three, there are beer and shot combos galore, a dangerously well-curated selection of craft spirits, and the best patty melt in the known world. In the summer, they also have the most epic backyard in the entire borough—and frozen Negronis.
A post shared by KARASU (@karasubk)
This not-so-secret speakeasy behind Walter Foods in Fort Greene features a potent mix of ultra-refined cocktails (try the Daydream made with white rum, nigori sake, green tea and coconut) and fresh takes on traditional Japanese izakaya fare. Their exquisite selection of sake, shochu and Japanese whisky make this a spot worth returning to time and time again.
68 Jay Street Bar
This no-frills bar in DUMBO is one of the few places that has managed to retain the feeling of the neighborhood as it once was: a home for artists and general eccentrics. With a welcoming mix of locals, hospitable bartenders and a straightforward, no-bullsh*t approach to drinks, this bar is the only place you should be imbibing in this neighborhood.
The Long Island Bar
Whether you're in the mood for an elegantly crafted cocktail (try their take on the Boulevardier), a perfectly poured beer or a bite to eat (it's all about the burger and the butterscotch pudding), the Long Island Bar is the place to go. It's like a '50s dinner interpreted through the lense of an art house auteur.
Bearded Lady
This bar is so laid back that you feel like you're on a remote Caribbean island rather than Washington Ave. Vibrantly hued, airy and light, Bearded Lady is the antithesis to the speakeasies of Manhattan. With creative, tiki-esque cocktails, epically delicious nachos and lounge-worthy tunes playing over the speakers, this watering hole is hard to leave.
Jack The Horse Tavern
This Brooklyn Heights stalwart mixes the coziness of an English pub with the fare of a French bistro and the decadence of the best cocktail lounges. Whether you're there to read the Sunday paper with a Bloody Mary or sip on Martinis at happy hour, this is the bar for you.
A post shared by Grand Army (@grandarmybar)
Grand Army
There are life-changing drinks at this bar—not to mention a truly spectacular shellfish platter. Approachable and inviting, with impeccable decor, this bar an absolute must-try Brooklyn destination. It's also one of the most Instagram-friendly bars in the borough.
Prime Meats
This timeless Brooklyn haunt has only been open since 2009, but it feels like it's been there for 100 years. Prime Meats is the type of establishment that you go for a quick drink and end up there for three hours. Whatever you do, don't leave without eating their burger—get it topped with Maytag blue cheese.
This dark, moody spot is the bar of choice for local bartenders. Between the Clover Club alums who work behind the bar, a slew of damn good beers on tap, and a dense selection of spirits on the backbar, you're guaranteed a good time at this underrated Gowanus gem.
Sunny's Bar
It's absolutely impossible to have a bad time at Sunny's. It doesn't matter if there's a hootenanny in the back room or if it's warm enough to enjoy the patio, this bar has an unparalleled warmth and an incredible cast of local characters. Step through the door and you'll feel like you've timewarped to another era and happened upon the best house party in town.
Bedford–Stuyvesant
Like Cheers for hip Brooklynites, this ultra-cozy Bed Stuy bar has a great selection of brews, seasoned bartenders slinging quality drinks, and a standout selection of tunes on their nightly playlists. And did we mention there's cake? Grab a slice of chocolate to enjoy with your Manhattan (or milk, if you prefer, they'll give it to you). And when it gets super frosty outside there's a fireplace to warm your mitts.
A post shared by Bar | 1057 Atlantic Ave, BKLYN (@diamondreef)
Diamond Reef
From the owners of Manhattan cocktail mecca Attaboy, this Brooklyn spot is the best and baddest, most delectably wonderful cocktail bar to hit the borough in years. Their cocktail menu manages to be at once approachable, fun and wildly inventive. Be sure to get the Penichillin', owner Sam Ross's frosty reimagining of his own original creation, the Penicillin.
At Blueprint, you'll find inventive, meticulously crafted cocktails that taste as good as they sound—try the Minotauro, a clarified milk punch with chai-spiced-infused cachaça, vanilla-infused Caribbean rum, fresh citrus and palm sugar. Inside, the bar has a dark but welcoming atmosphere and delicious bar snacks (think cheese plates and charcuterie rather than nachos and pretzels). If that doesn't win you over, there also happens to be a lovely outdoor patio.
King Tai
With an ever-evolving cocktail menu that features exotic spirits and tropical-inspired flavors, a neon-drenched interior, and an impressive roster of DJs, King Tai is the place to be in Crown Heights. There's nowhere else in the neighborhood where you're guaranteed to shake your ass harder—or more joyously—than here. Afterwards, take your sweaty, blissed-out self around the corner to Chris, where you'll find some seriously good Jamaican food to satisfy that 4 a.m. hunger.
Bar Chord
If you like live music (especially rock n' roll), this Cortelyou Road hideaway is a must-visit venue. Between the classic guitars that adorn the walls and the nightly bands that play in the front room, this bar wears its love of music on its proverbial sleeve. With a giant outdoor courtyard, great brews and cocktails, there's a little something for everyone.
Church Bar
The best way to describe this quaint Kensington cocktail bar is no-frills, no-fuss, and no-bullsh*t. Come for the delicious cocktails, hospitable and well-versed staff, chill vibes and friendly locals. If you moved to the neighborhood, you'd be drinking here everyday too.
A post shared by Erv's on Beekman (@ervsonbeekman)
Prospect-Lefferts Garden
Erv's On Beekman
The first time we stumbled into Erv's, we were instantly stoked about our accidental discovery. Inside, it's full of boisterous locals with excellent jams playing on the stereo, and each drink is better than the last. It's always hard to leave, and when you do, you'll be frustrated that you don't live closer.
Not to be confused with the iconic gay bar in Crown Heights, the other, more remote Starlite Lounge is the perfect dive bar to visit if you happen to be looking for a watering hole in Midwood. This isn't a place to drink cocktails (or even ask for them), but if you want cheap liquor and beer, and a game of darts or pool, look no further.
Irish Haven
This old-timey Irish bar keeps things real. A night at this locals-only haunt is guaranteed to be a strange, drunken adventure. Just don't get yourself into a brawl by spouting off at the mouth—they don't call 'em the fightin' Irish for nothing.
The Owl's Head
Opened in 2011, this Bay Ridge wine bar is a neighborhood gem. Owner John Avelluto's menu features some of the most eclectic, diverse and delicious biodynamic wines from around the world (seriously, this list is on par with some of the best in Manhattan). There's even a house red that's made in collaboration with Red Hook Winery.
Ruby's Bar & Grill
You can't go to the Coney Island Boardwalk without stopping for a drink at Ruby's. This beachside bar is as much of a tradition as Nathan's hotdogs or the Cyclone rollercoaster. Snag a table on the boardwalk or belly up to the bar and order a pitcher of beer (a selection from the nearby Coney Island Brewing Co. would be appropriate) or a frozen drink, and take in some of the best people watching known to Brooklyn.

redpajama_set_name: RedPajamaCommonCrawl


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Reach out to your representatives
One of the most effective ways to make change is to contact your representatives and tell them what's important to you. Reach out to your representatives today to let them know you want a ban on lethal autonomous weapons.
Live in the US? Find your representatives here.
Live in the UK? Find your Members of Parliament here.
Live in Canada? Find your Members of Parliament here.
Live in Europe? Find members of the European Parliament here.
Trying to figure out what to say?
Find inspiration from some of the statements below by the organizations seeking to ban autonomous weapons:
Lethal autonomous weapons systems, or killer robots, would be able to select and attack targets without human control, delegating decisions over whom to attack and kill, to a machine. Although fully autonomous weapons are not yet in operation, a number of countries are developing capacities for combat machines that have greater autonomy – including the UK, China, Israel, Russia and the United States. ... Action by states is needed now to develop an understanding over what is unacceptable when it comes to the use of autonomous weapons systems, and to prohibit these activities and development through an international treaty.
The Future of Life Institute
Autonomous weapons select and engage targets without human intervention. They might include, for example, armed quadcopters that can search for and eliminate people meeting certain pre-defined criteria, but do not include cruise missiles or remotely piloted drones for which humans make all targeting decisions. Artificial Intelligence (AI) technology has reached a point where the deployment of such systems is — practically if not legally — feasible within years, not decades, and the stakes are high: autonomous weapons have been described as the third revolution in warfare, after gunpowder and nuclear arms.
Starting a military AI arms race is a bad idea, and should be prevented by a ban on offensive autonomous weapons beyond meaningful human control.
The development of fully autonomous weapons—"killer robots"—that could select and engage targets without human intervention need to be stopped to prevent a future of warfare and policing outside of human control and responsibility. Human Rights Watch investigates these and other problematic weapons systems and works to develop and monitor international standards to protect civilians from armed violence.
International Committee for Robot Arms Control
Given the rapid pace of development of military robotics and the pressing dangers that these pose to peace and international security and to civilians in conflict, we call upon the international community for a legally binding treaty to prohibit the development, testing, production and use of autonomous weapon systems in all circumstances.
Machines should not be delegated with the decision to kill or use violent force.


Synopsis: THE AFTERMATH is set in postwar Germany in 1946. Rachael Morgan (Keira Knightley) arrives in the ruins of Hamburg in the bitter winter, to be reunited with her husband Lewis (Jason Clarke), a British colonel charged with rebuilding the shattered city. But as they set off for their new home, Rachael is stunned to discover that Lewis has made an unexpected decision: They will be sharing the grand house with its previous owners, a German widower (Alexander Skarsgård) and his troubled daughter. In this charged atmosphere, enmity and grief give way to passion and betrayal.
Synopsis: Earning him a Best Actor Oscar® nomination, Willem Dafoe is mesmerising as legendary painter Vincent van Gogh in At Eternity's Gate. Covering his time in Arles and Auvers-sur-Oise, we find a complex, troubled soul who is seeking a new visual language while also struggling to make sense of himself and his connection to those around him, including friend and fellow artist Paul Gauguin (Oscar Isaac) and a confiding priest (Mads Mikkelsen). Shot with a painter's eye, Julian Schnabel once again shows his talent for working with complex biography and the lives of artists as in The Diving Bell and the Butterfly and Basquiat.
Synopsis: A young man of just 22, dressed in a black beret and a dark narrow suit, is on an aeroplane flying from St Petersburg to Paris. It is 1961 and Rudolf Nureyev, not yet the imperious figure of legend, is a member of the world-renowned Kirov Ballet Company, travelling for the first time outside the Soviet Union. Parisian life delights Nureyev and the young dancer is eager to consume all the culture, art and music the dazzling city has to offer. But the KGB officers who watch his every move become increasingly suspicious of his behaviour and his friendship with the young Parisienne Clara Saint. When they finally confront Nureyev with a shocking demand, he is forced to make a heart-breaking decision, one that may change the course of his life forever and put his family and friends in terrible danger. From Nureyev's poverty-stricken childhood in the Soviet city of Ufa, to his blossoming as a student dancer in Leningrad, to his arrival at the epicentre of western culture in Paris in the early 1960s and a nail-biting stand-off at the Le Bourget airport, "THE WHITE CROW" is the true story of an incredible journey by a unique artist who transformed the world of ballet forever.


My blog post to-do list is growing. One of them I've made a start on, but I'm really struggling to write it. Most of that is because I'm now taking the domperidone three times a day and it makes me really dopey. I can cope with it- it's not the worst side effect I've had to deal with by a long stretch, but it does mean I have very little brain power for things like writing blog posts that aren't just a stream of consciousness.
The dopeyness has meant I've not been up to much. I've been on my computer everyday but I've not even logged into World of Warcraft for a bit as it's just too much. I've spent a lot of time watching kittehs and some watching penguins. I have caught up with Total Biscuit's Hearthstone videos finally, which is something. I love his gimmick decks as they're really entertaining, and even if my speech processing is poor (though TB's voice is easier than most for me to understand) I can at least see what's happening for myself.
I've had no speech myself for a few days. It's starting to get annoying. The new AAC app is really helping with that, though when I'm on my computer it's a little awkward to use as I don't have speakers connected to it at the moment. What I really need to do is get the front headphone jack hooked up so I can plug my headphones in there, my speakers hooked up again and then figure out having the different programs going out of different audio outputs. There's got to be an easy way to do it but my brain is too bleh to figure it out (and I don't have the speakers in here at the moment).
I accidentally wiped the data partition on my tablet while trying to update the Google Apps on it (I have them integrated into the ROM). Since I'd done that and Paranoid Android is currently in the process of reworking a lot of its features, I took the opportunity to try a different ROM out. I'm now on PAC-man, which I tried before but had issues with at the time. This time it seems to be working okay despite me being on the nightlies, and I really like some of the extra features. I also changed my font to Comic which made Johan facepalm but reminds me of Doge so I'm happy.
I can't remember if I mentioned it here, but I discovered Johan had set the spam filter on our email server to automatically reject spam, and that it was definitely having some false positives of my email. Included in that was an email from EE, so there's a good chance they actually responded to my email to them in December and it got eaten. Because of this, I've moved over to Google Apps for my email. Right now I'm importing all my emails from my other GMail account so I can have the few I still need to reply to in the right place. Since I basically archived all my emails from 2009 it's going to take a while 😛 I'm hoping it'll be done by tomorrow. As I still have them all on the other account, I'll be deleting all but the recent ones to try and keep this one a bit clearer. I wish there was an easy way to just import the recent emails, but forwarding screws up the replying (I tried) and there's no date selection tool that I found. Some of the emails I've seen are from when I was living alone, and it really reminded me about how badly that went. I know I can't manage living independently, but it feels weird to see emails about how I would forget to eat and stuff. Not that it's much improved now, but at least with the two of us one of us has a chance to remember.
Primrose, my new microwaveable penguin, is fitting in well. She's not quite as silly as the other penguins but I'm sure she'll learn. She was disapproving of me changing my font on my tablet to Comic so I need to give her an education on Doge and similar things. She does help me with pain relief along with Penguin, and Penny and Po keep me company while they're in the kitchen. Penelope wants a wheelchair so I am needing to find a penguin sized one (she thinks they're cool).
Johan went out on Saturday to see some friends we'd made on Twitter who happen to be local. From what he told me it went well and he brought me back cake, which was amazing. Since Johan wasn't too far away I was okay with being left, as he had money for a taxi if an emergency did crop up. With my carer coming though it was fine. He has discovered that if he uses his inhaler he can manage around cats much better. Turns out treating asthma helps 😉 I'm trying to convince him to take his preventative inhaler twice a day as he's meant to, so hopefully he will get less symptoms as that builds up in his system.


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Georgian Bay General Hospital completes major renovation in sterilization department
As part of its commitment to quality and safety, Georgian Bay General Hospital (GBGH) has completed a critical and complex renovation project in its medical device reprocessing department (MDRD). Within the hospital, this is where instruments, devices and equipment are sterilized to meet the highest standards for quality and safety. MDRD staff and project team members gathered in part of the new space to celebrate this milestone for GBGH and its surgical program. The hospital plans for continued growth in its surgical department and other services to align with the evolving needs of the community it serves.
December 14, 2022 – Georgian Bay General Hospital (GBGH) is celebrating the completion of a major renovation to its sterilization department, which took place in two phases over the past four years amid pandemic-related delays. The completion of this project marks a major milestone for the hospital which relies on safe, high-quality sterilization for a variety of instruments, equipment and devices used throughout the hospital.
"The redevelopment of our MDRD is an essential part of GBGH's strategic objectives to grow our surgical program to align our services with community needs, bringing care closer to home," says Matthew Lawson, president and CEO, GBGH. "Over the past six months, GBGH has been able to increase surgical volumes to 163 per cent of pre-pandemic levels. We've been successful in our strategy to grow our surgical volumes by investing in equipment, broadening the range services we provide and optimizing our OR spaces. Finalizing our new state-of-the-art MDRD project perfectly complements our hospital's ongoing surgical expansion plans."
The two-phase project began in 2018, after GBGH proactively initiated an assessment of its sterilization processes and physical space to identify areas to improve efficiency and update equipment. After receiving recommendations, the hospital had to outsource medical device reprocessing to an external supplier, enabling a renovation project of this size to begin. Phase 1 of the project took place pre-pandemic and then was delayed. Phase 2 work has since occurred over the past eight months, resulting in the final completion this month.
Due to its complexity and duration, this project required great patience and flexibility among MDRD staff and project team members. Recognizing the vast improvements the redevelopment would have on the standard of device sterilization at GBGH, the team has embraced learning the latest equipment and the changes in workflow.
With the completion of this project, GBGH will no longer be externally contracting sterilization, which consisted of transporting back and forth to the Greater Toronto Area the devices, instruments and equipment the hospital uses daily. Bringing sterilization back in-house is a cost savings for the hospital and reduces the likelihood of logistical issues and errors with transport.
Between April 2021 and March 2022, GBGH performed 4,012 surgical procedures despite the pandemic-related ramp down of non-urgent and elective procedures. The hospital's plans to continue growing its surgical program are in part due to more than 3,700 local patients travelling elsewhere for surgical procedures which can be performed at GBGH. This equates to more than 600,000 kilometres and 7,500 hours of travel that could have been saved by local patients.
Director, Communications & Stakeholder Relations
moorejen@gbgh.on.ca
More About GBGH


The training will conclusively handle issues to do with the understanding gender, gender awareness and advocacy, human rights, and gender-based violence, gender analysis, integration and gender mainstreaming, leadership and governance and gender budgeting.
The training is meant for professionals working in the all the three sectors of society including the Public Sector, Government Ministries and State Corporations; Civil Society Organizations (NGOs, FBOs and CBOs); and the Private sector.
To increase participants' knowledge on gender-responsive budgeting.


Tellurium Automated Testing Framework is an open source automated testing framework for testing web applications. Tellurium evolved from Selenium framework about 2 years ago with a different testing approach. Tellurium is built on UI module concept, which makes it possible to write reusable and easy to maintain tests against the dynamic RIA based web applications. UI module is collection of UI(DOM) elements grouped together. Current version is 0.8.0.
The easiest way to create a Tellurium project is to use Tellurium Maven archetypes.
Create the project structure using Tellurium Maven archetype.
Create the reusable methods for different operations on the UI module.
Create test cases in Java, Groovy or DSL. Tellurium supports JUnit/TestNG/easyb test cases.
Most existing web testing frameworks, like Selenium, primarily focus on individual UI elements such as links and buttons, Tellurium on the other hand, groups UI elements as UI objects into UI module.
Here we are defining that the UI consists of one Input textbox element and two submit buttons.
Adoption of UI module makes Tellurium expressive and easy to understand in the context of tests. Tellurium sets the Object to Locator Mapping at runtime using the attributes from the composite locator(clocator). This makes Tellurium more robust and responsive to changes from internal UI elements.
issueResult.header returns the first column in the header row which is ID column.
issueResult returns the cell for second row and second column.
Tellurium API provides with various methods to access and manipulate the web data. Tellurium enforces clear separation between the UI and the test code. In an agile world where UI changes rapidly, having this clear separation makes it easier to modify UI with minimal changes to the test code.
Tellurium started as small core project but quickly expanded into multiple sub-projects.
Tellurium Core: UI module, APIs, DSL, Object to Runtime Locator mapping, test support.
Tellurium Engine: Based on Selenium core with UI module, CSS selector support, macro command and exception hierarchy support.
Tellurium UI Module Plugin(Trump): Firefox plugin that automatically generates the UI module after user selects the UI elements on the web page being tested. Trump validates the UI module by evaluating each UI elements attributes, then generates the runtime locator and verifies the locators are valid.
Tellurium-IDE: Firefox plugin that QA group and non programmers can use to record user actions on the web page and replay them. Plugin automatically generates Tellurium commands and UI modules in DSL script format. The DSL script can be exported as Groovy script. Replaying the script is done by the built in test runner.
TelluriumWorks: Java Swing application to edit and run Tellurium-IDE generated test scripts.
Tellurium works in two modes. First mode works as a wrapper to the Selenium framework. Tellurium core generates the runtime locator based on the attributes defined in clocator of the UI module. The generated locator is then passed in Selenium calls to the Selenium core with Tellurium extensions.
Second mode uses Tellurium Engine. Tellurium Core will convert the UI module into JSON notation and passes it to the Tellurium Engine when the UI module is used first time. Engine uses the Santa algorithm to locate the whole UI module and caches it. For subsequent calls, the cached UI module will be used instead of locating again, thereby improving the test execution speed.
Uses abstract UI objects to encapsulate web UI elements such as InputButton, Selector, UrlLink, List, Table, Frame.
UI templates and UID Description Language(UDL) are used to represent UI elements like HTML Tables or List where the content changes dynamically.
Internationalization(i18n) support for Strings and exception messages.
CSS selectors are supported to improve test speed in IE.
Locator caching and command bundling further improve the test speed.


text:
Chippewa Herald
Dunn County News
From the files: Dunn County slot machines destroyed in 1944
From the files
SOFI DOANE For Dunn County News
Wednesday, Jan. 27, 1982: It's the dog days of winter. A dog rambles on snow-covered Lake Menomin while the winter sun tries to melt some of the snow which surrounds the ice fishing huts.
CONTRIBUTED, Dunn County Historical Society
Saturday, Feb. 23, 1884
The narrow escape of the Central school building from destruction last week, recalls to the minds of the older residents of the city the fact that this community had previously lost a school building by fire.
In 1861 the town bought the two lots that now constitute the northeast corner of the Central School block. On this, in 1862, fronting on Main street, was erected a one-story building, 30x40 feet, with a belfry, at a cost of $1,500.
In November, 1867, one evening after a religious meeting had been held therein, the building was discovered to be on fire, and was destroyed in spite of the utmost efforts on the part of the citizens to save it.
The only other buildings then on that street, were a carpenter shop owned by Wm. Huber, standing in the rear of the location of his present brick block, Mrs. Fowler's old store, which was only partly completed, and the Vance House which occupied the ground where now stands Perrault's Union Hotel.
Huber's shop being the only available building, was at once secured and fitted up for school purposes. This was occupied until 1869, when the new building was completed at an expense of $10,000.
Friday, Feb. 23, 1894
Those of our citizens so inclined have been entertained this week with a series of lectures on and exhibitions of hypnotism at the opera house by Prof. Roche, who is said to be the peer of Bishop and Johnstone, who have in times past produced sensations in this line in different portions of the country.
Prof. Roche describes hypnotism as a science, having for a basic principle, concentration of thought.
Given a subject with powers of concentration, whose mind is passive, and the professor really performs a number of acts quite wonderful to the layman.
For instance, by a few passes of his hands and apparently by a powerful effort of the will he puts the subject into an hypnotic condition wherein there is speechlessness, loss of memory, loss of power of locomotion, rigidity of muscles, or imaginary scenes become real as life, or an hypnotic sleep is produced wherein the subject is insensible to pain—as he chooses.
A specimen of mind-reading was given Thursday. At the noon hour Prof. Roche, carefully blindfolded, accompanied by a committee composed of G. L. Hatter, Geo. Farnsworth and S. L. Heafield, starting from the court house, drove a team at break-neck speed through Main and Broadway to the Hotel Royal.
There, still securely blindfolded, the professor proceeded straight to the register and without hesitation marked with a pencil a name the committee had previously selected. It was an exceedingly interesting experiment and was witnessed by a large number of our citizens.
Thursday, Feb. 20, 1919
A Stout Institute graduate has been made Chief Dietitian at a Mammoth, New York hospital.
Appointed chief dietitian at one of Uncle Sam's largest and most modern convalescent hospitals is the honor which has been conferred by Washington officials on Winnifred Short of Superior, a graduate of Stout Institute with the class of 1913, who is well-known to many Menomonie people.
The Superior Telegram recently published the following article concerning Miss Short and her work:
Miss Short, known as "Winnie" or "Shorty" by her intimate friends and hundreds of Superior High school students, is short in stature but positively towering in executive ability.
When she received her appointment from the Red Cross as army dietitian and was sent to Colonia, N. J.. to organize the diet kitchen at the Colonia hospital. Miss Short set out with all the aplomb of a seasoned veteran of army work.
Of course she was well prepared for her new duties for in addition to her training at Stout institute and experience as domestic science instructor at the Superior High school, she had taken a postgraduate course at Columbia University and a special course in dietetics at the Battle Creek sanitarium.
When the U. S. Debarkation hospital at New York was built Miss Short was selected from a number of expert dietitians for the position of chief dietitian which requires great ability.
The hospital is known as the Grand Central Palace. It is a building of thirteen stories, occupying an entire city block, and containing 4,500 beds. A staff of 750 people, doctors, nurses and attendants is required. It is said to be the most up-to-date institution of its kind in America,
Wednesday, Feb. 23, 1944
Three 5-cent slot machines, often referred to by the public as "one-armed bandits," met their doom on the concrete driveway leading into the county jail shortly after noon Tuesday when Sheriff Fred Einum dealt a series of devastating blows with a hammer that smashed the mechanical money makers to smithereens.
The machines, confiscated by the sheriff late Monday afternoon at the Lois Kewin tavern, off of STH 64, in the Town of Otter Creek, will neither "pay out" or "take in" anymore—for they have "eaten" their last buffalo nickels, or other nickels that are faced with other historic figures.
Dist. Atty. Clarence E. Smith received a complaint that slot machines were being operated in the Kewin tavern and so he issued search warrant with the approval of the county court, and Sheriff Einum, armed with the warrant, made a visit to the tavern and found the three machines in a back room.
He took them, and directed Mrs. Kewin into county court where she pled guilty to the charge of being in possession of a gambling apparatus.
Judge Peterson fined her $25, and ordered the sheriff to smash the machines in the presence of at least one representative of the press, and turn the money over to Clerk of Court Deloris Brown.
As far as the sheriff knows, Dunn County is now free of slot machines.
The groundwork on which future decisions will be based was laid Tuesday night when the Menomonie board of education met.
Discussion centered on district-owned versus contract carrier transportation.
Supt. William Terrill reported the district should either purchase a new bus or have a contract carrier transport students.
Figures presented indicate the cost would be approximately the same. The district is now operating a 1963 model bus.
Cost of a new unit is estimated at $8,200 and deducting the federal tax and trade-in gives a cost of $6,000. The cost of operating a bus by the district is $4,286 annually with major expenses being $1,000 for depreciation, $1,175 for drivers salary, $650 for gas, oil and grease, and $570 for labor and repair parts.
Contract carrier service cost $4,262 annually in the city. The cost is $12.50 per day for the bus, $5 for salary and $6.18 for mileage. The board decided to table the matter for a month to allow a study of the cost figures.
With President Theodore Bissell casting the deciding vote, the Boyceville Village Board Monday night approved a Municipal Hall improvement project costing some $2,500.
The hall houses the public library and a restroom in this section of the building will be converted into a handicapped accessible facility. This work will cost about $1,900.
Additionally, two upper level windows where frames have rotted will be replaced for $600. Mclntyre Construction of Boyceville will do the work.
Joining Bisseil in supporting the project were Trustees Gilbert Krueger, Arthur Nelson Jr. and Gilbert Sykora. Opposed were Robert Marlett and Alien Froiland; Maynard Pittman was absent.
The projects were forwarded by Sykora, who represents trustees on the Library Board. He said the original estimate for making the restroom accessible to the handicapped was $1,203.
However, he continued, the cost jumped to $1,900 because it was decided to replace the floor and ceiling. Replacing the windows was endorsed by Bissell. "If we are going to keep the building, we have to do something with the windows," he declared.
Besides the building improvements, the board voted to install an electronic air cleaner on the furnace that serves the library. This project, too, was forwarded by Sykora who said the librarian suffers from allergies.
"I think a number of people suffer from this type allergy," he added. Estimated cost of this work is $745.
In other action, the board awarded an $11,200 contract to Century Fence to install new fencing at Pafko Park.
The Menomonie Police Department has embarked on a program to have trained chaplains work side-by-side with police officers.
Sgt. Rick Hollister, who is the chaplain coordinator for the department said this is a first for Menomonie.
"The chaplain program will be a huge resource for the department and community," Hollister said.
Thus far, four chaplains have been sworn in. They are all pastors and include David Whelan of Peace Lutheran Church, Geoff Scott of Christ Lutheran Church, Marilynn Rushton of United Methodist Church and Rolf Morck of Christ Lutheran. All four pastors have gone through the department's Citizens academy, have three years experience in the ministry, are ordained pastors and are familiar faces in the community.
"The Chaplaincy Program will strive to enhance the Menomonie Police Department as a caring institution," Hollister said.
The program has already proven its worth. When a motorist recently died at the I-94 rest stop, the police officer called ahead and a chaplain was at Myrtle Werth Hospital when the family arrived. The chaplain was able to comfort and assist the family, and they were most grateful for the assistance.
Menomonie's Army National Guard unit, Alpha Company of the 1-128th Infantry, is expected to depart on Monday, Feb. 23. The company will travel to Fort Bliss, Texas, for two to three months of training prior to being deployed to Iraq for the second time.
To show support along the route Company A will be taking as they leave town, residents and businesses are being asked to fly flags and display signs- bidding them farewell and good luck-before and on the day of the unit's departure.
Company A will join an approximate 3,200 Wisconsin Army National Guard soldiers deploying with the 32nd Infantry Brigade Combat team for a tour of duty in support of Operation Iraqi Freedom.
According to Maj. Jackie Guthrie of the Wisconsin National Guard public affairs office, units will travel from their local armories to Ft. Bliss, Texas, where they will spend about two months in mobilization training before heading to Iraq. The expected term of service in the current deployment is expected to be about one year.
Sunday, Feb. 23, 2014
The heavy wet snow that turned the western Wisconsin into a winter wonderland was anything but wonderful.
The system hit around mid afternoon Thursday and continued until well into the evening in most parts of Dunn County.
Heeding National Weather Service warnings about the storm, school kids in Dunn County were sent home around mid-day on Thursday and evening activities were cancelled.
Yet another snow day was declared by the Boyceville and Colfax school districts on Friday, while Menomonie was already scheduled to be off and Elk Mound began its mid-winter break.
The weight of the white the stuff took its toll on the roof of O'Reilly's Auto Parts Store in North Menomonie. Shortly before 9:30 p.m Thursday, two employees were closing up shop lat when a load-bearing beam gave way and a portion of the roof caved in.
According to Menomonie Fire Department Battalion Chief John Entorf, "The roof fell about three feet before it caught on some shelving." No one was injured, and there was no damage to water or gas lines as a result.
Sofi Doane
Sofi Doane is the Collections Manager for the Dunn County Historical Society. She can be reached at 715-232-8685 or dchs@dunnhistory.org
Winnifred Short
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Steve Robertson
Back to Directors
Steve Robertson has nearly three decades experience in exploration and mine operations in British Columbia. In November 2017 Steve joined Sun Metals as President & CEO, from Imperial Metals where he spent the previous 24 years. Steve earned his BSc in geology from University of Alberta in 1989 and the bulk of his career has been spent in exploration.
Steve has always been actively engaged in the mining community. He previously served as director for AME, sat on the Geoscience BC Technical Advisory Committee and was a founding director of the BC Aboriginal Mine Training Association. In 2016 Steve was awarded the E.A. Scholz Award for Excellence in Mine Development for his leadership in development of the Red Chris mine.
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text:
\section{references}

\defDepartment of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel{Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel}
\def\Conc{Department of Mathematics and Statistics, Concordia
University, Montreal, Quebec, Canada}
\author{ Hershy Kisilevsky}
\address{\Conc}
\email{kisilev@mathstat.concordia.ca}

\author{ Danny Neftin}
\address{Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel}
\email{neftind@tx.technion.ac.il}%

\author{ Jack Sonn }
\address{Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel}
\email{sonn@math.technion.ac.il}

\begin{abstract}
It is known
(\cite{KS}, \cite{N})
that for any prime $p$ and any finite semiabelian $p$-group $G$, there exists a (tame) realization
of $G$ as a Galois group over the rationals $\Bbb Q$ with exactly $d=\dg(G)$ ramified primes, where
$\dg(G)$ is the minimal number of generators of $G$, which solves the minimal ramification problem
for finite semiabelian $p$-groups.  We generalize this result to obtain a theorem on finite
semiabelian groups and derive the solution to the minimal ramification problem for a certain family
of semiabelian groups that includes all finite nilpotent semiabelian groups $G$.  Finally, we give some indication of
the depth of the minimal ramification problem for semiabelian groups not covered by our theorem.
\end{abstract}

\maketitle

\section{Introduction}

Let $G$ be a finite group.  Let $d=\dg(G)$ be the smallest number for which there exists a subset $S$
of $G$ with $d$ elements such that the normal subgroup of $G$ generated by $S$ is all of $G$.  One
observes that if $G$ is realizable as a Galois group $G(K/\mathbb{Q})$ with $K/\mathbb{Q}$ tamely ramified (e.g.
if none of the ramified primes divide the order of $G$), then at least $\dg(G)$ rational primes
ramify in $K$ (see e.g. \cite{KS}).  The \it minimal ramification problem for $G$ \rm is to realize
$G$ as the Galois group of a tamely ramified extension $K/\mathbb{Q}$ in which exactly $\dg(G)$ rational
primes ramify. This variant of the inverse Galois problem is open even for $p$-groups, and no
counterexample has been found.  It is known that the problem has an affirmative solution for
all semiabelian $p$-groups, for all rational primes $p$ (\cite{KS},\cite{N}).  A finite
group $G$ is \it semiabelian \rm if and only if $G \in \mathcal{S}\mathcal{A}$, where $\mathcal{S}\mathcal{A}$ is the smallest
family of finite groups satisfying: (i) every finite abelian group belongs to $\mathcal{S}\mathcal{A}$. (ii) if $G
\in \mathcal{S}\mathcal{A}$ and $A$ is finite abelian, then any semidirect product $A \rtimes G$ belongs to
$\mathcal{S}\mathcal{A}$. (iii) if $G \in \mathcal{S}\mathcal{A}$, then every homomorphic image of $G$ belongs to $\mathcal{S}\mathcal{A}$. In this
paper we generalize this result to arbitrary finite semiabelian groups by means of a ``wreath
product length" $\wl(G)$ of a finite semiabelian group $G$.  When a finite semiabelian group $G$ is
nilpotent, $\wl(G)=\dg(G)$, which for nilpotent groups $G$ equals the (more familiar) minimal number
of generators of $G$.  Thus the general result does not solve the minimal ramification problem for
all finite semiabelian groups, but does specialize to an affirmative solution to the minimal
ramification problem for nilpotent semiabelian groups. Note that for a nilpotent group $G$, $\dg(G)$ is $max_{p||G|}\dg(G_p)$ and not $\sum_{p||G|}\dg(G_p)$, where $G_p$ is the $p$-Sylow subgroup of $G$. Thus, a solution to the minimal ramification problem for nilpotent groups does not follow trivially from the solution for $p$-groups.

\section{Properties of wreath products}

\subsection{Functoriality}
The family of semiabelian groups can also be defined using wreath
products. Let us recall the definition of a wreath product. Here and
throughout the text the actions of groups on sets are all right
actions. \begin{defn} Let $G$ and $H$ be two groups that act on the
sets $X$ and $Y$, respectively. The \it (permutational) wreath
product \rm $H\wr_X G$ is the set $H^X\times G=\{(f,g)|f:X\rightarrow H,g\in
G\}$ which is a group with respect to the multiplication: $$
(f_1,g_1)(f_2,g_2)=(f_1f_2^{g_1^{-1}},g_1g_2),$$ where
$f_2^{g_1^{-1}}$ is defined by $f_2^{g_1^{-1}}(x)=f_2(xg_1)$ for any
$g_1,g_2\in G,x\in X,f_1,f_2:X\rightarrow H$. The group $H\wr_X G$ acts on
the set $Y\times X$ by $(y,x)\cdot (f,g)=(yf(x),xg)$, for any $y\in
Y,x\in X,f:X\rightarrow H,g\in G$. \end{defn}
\begin{defn}The \textit{standard (or regular) wreath product} $H\wr G$ is defined
as the permutational wreath product with $X=G$,$Y=H$ and the right regular actions. \end{defn}

The functoriality of the arguments of a wreath product will play an important
role in the sequel. The following five lemmas are devoted to these functoriality properties.

\begin{defn} Let $G$ be a group that acts on $X$ and $Y$. A map
$\phi:X\rightarrow Y$ is called a $G$-map if $\phi(xg)=\phi(x)g$ for every
$g\in G$ and $x\in X$. \end{defn} Note that for such $\phi$, we also
have
$\phi^{-1}(y)g=\{xg|\phi(x)=y\}=\{x'|\phi(x'g^{-1})=y\}=\{x'|\phi(x')=yg\}=\phi^{-1}(yg).$
\begin{lem}\label{G-map induces an epimophism} Let $G$ be a group
that acts on the finite sets $X,Y$ and let $A$ be an abelian group.
Then every $G$-map $\phi:X\rightarrow Y$ induces a homomorphism $\widetilde{\phi}:
A\wr_X G\rightarrow A\wr_Y G$ by defining: $(\widetilde{\phi}(f,g)) = (\hat{\phi}(f),g)$
for every $f:X\rightarrow A$ and $g\in G$, where $\hat{\phi}(f):Y\rightarrow A$ is
defined by: $$\hat{\phi}(f)(y)=\prod_{x\in \phi^{-1}(y)}f(x),$$ for
every $y\in Y$. Furthermore, if $\phi$ is surjective then $\widetilde{\phi}$ is
an epimorphism. \end{lem} \begin{proof} Let us show the above $\widetilde{\phi}$
is indeed a homomorphism. For this we claim:
$\widetilde{\phi}((f_1,g_1)(f_2,g_2))=\widetilde{\phi}(f_1,g_1)\widetilde{\phi}(f_2,g_2)$ for every
$g_1,g_2\in G$ and $f_1,f_2:X\rightarrow A$. By definition:
\begin{equation*}
\widetilde{\phi}(f_1,g_1)\widetilde{\phi}(f_2,g_2)=(\hat{\phi}(f_1),g_1)(\hat{\phi}(f_2),g_2)=(\hat{\phi}(f_1)\hat{\phi}(f_2)^{g_1^{-1}},g_1g_2),\end{equation*}
while:
$\widetilde{\phi}((f_1,g_1)(f_2,g_2))=\widetilde{\phi}(f_1f_2^{g_1^{-1}},g_1g_2)=(\hat{\phi}(f_1f_2^{g_1^{-1}}),g_1g_2).$
We shall show that $\hat{\phi}(f_1f_2)=\hat{\phi}(f_1)\hat{\phi}(f_2)$ and
$\hat{\phi}(f^g)=\hat{\phi}(f)^g$ for every $f_1,f_2,f:X \rightarrow A$ and $g\in G$.
Clearly this will imply the claim. The first assertion follows
since: \begin{equation*}\hat{\phi}(f_1f_2)(y)=\prod_{x\in \phi^{-1}(y)}
f_1(x)f_2(x)=\prod_{x\in \phi^{-1}(y)} f_1(x)\prod_{x\in
\phi^{-1}(y)} f_2(x)=\end{equation*}$$ =
\hat{\phi}(f_1)(y)\hat{\phi}(f_2)(y).$$ As to the second assertion we have:
\begin{equation*} \hat{\phi}(f^g)(y)= \prod_{x\in \phi^{-1}(y)}
f^g(x)=\prod_{x\in \phi^{-1}(y)} f(xg^{-1})
=\end{equation*}\begin{equation*} = \prod_{x'g\in \phi^{-1}(y)}
f(x')  = \prod_{x'\in \phi^{-1}(y)g^{-1}} f(x').\end{equation*}
Since $\phi$ is a $G$-map we have
$\phi^{-1}(y)g^{-1}=\phi^{-1}(yg^{-1})$ and thus \begin{equation*}
\hat{\phi}(f^g)(y)= \prod_{x\in \phi^{-1}(y)g^{-1}} f(x)= \prod_{x\in
\phi^{-1}(yg^{-1})} f(x) =  \hat{\phi}(f)^g(y). \end{equation*} This
proves the second assertion and hence the claim. It is left to show
that if $\phi$ is surjective then $\widetilde{\phi}$ is surjective. Let $f':Y\rightarrow
A$ and $g'\in G$. Let us define an $f:X\rightarrow A$ that will map to $f'$.
For every $y\in Y$ choose an element $x_y\in X$ for which
$\phi(x_y)=y$ and define $f(x_y):=f'(y)$. Define $f(x)=1$ for any
$x\not\in\{x_y|y\in Y\}$. Then clearly
$$\hat{\phi}(f)(y)=\prod_{x\in\phi^{-1}(y)}f(x)=f(x_y)=f'(y).$$ Thus,
$\widetilde{\phi}(f,g')=(\hat{\phi}(f),g')=(f',g')$ and $\widetilde{\phi}$ is onto. \end{proof}
\begin{lem}\label{construction of G-maps} Let $B$ and $C$ be two
groups. Then there is a surjective $B\wr C$-map
$\phi:B\wr C\rightarrow B\times C$ defined by: $\phi(f,c)=(f(1),c)$ for
every $f:C\rightarrow B, c\in C$. \end{lem} \begin{proof} Let
$(f,c),(f',c')$ be two elements of $B\wr C$. We check that
$\phi((f,c)(f',c'))=\phi(f,c)(f',c')$. Indeed, \begin{equation*}
\phi((f,c)(f',c'))=\phi(ff'^{c^{-1}},cc')=(f(1)f'^{c^{-1}}(1),cc')=(f(1)f'(c),cc')=\end{equation*}
$$ =(f(1),c)(f',c)=\phi(f,c)(f',c'). $$ Note that the map
$\phi$ is surjective: For every $b\in B$ and $c\in C$, one can
choose a function $f_b:C\rightarrow B$ for which $f_b(1)=b$. One has:
$\phi(f_b,c)=(b,c)$. \end{proof}

The following Lemma appears in \cite[Part I, Chapter I, Theorem 4.13]{B} and describes
the functoriality of the first argument in the wreath product.
\begin{lem}\label{functoriality of
first argument} Let $G,A,B$ be groups and $h:A\rightarrow B$ a homomorphism (resp. epimorphism).
Then there is a naturally induced homomorphism (resp. epimorphism) $h_*:A\wr G\rightarrow B\wr
G$ given by $h_*(f,g)=(h\circ f,g)$ for every $g\in G$ and $f:G\rightarrow
A$. \end{lem}

The functoriality of the second argument is given in \cite[Lemma 2.15]{N} whenever the first argument is abelian:
\begin{lem}\label{functoriality with
second argument} Let $A$ be an abelian group and let $\psi:G\rightarrow H$
be a homomorphism (resp. epimorphism) of finite groups. Then there is a homomorphism (resp. epimorphism)
$\widetilde{\psi}:A\wr G\rightarrow A\wr H$ that is defined by:
$\widetilde{\psi}(f,g)=(\hat{\psi}(f),\psi(g))$ with $\hat{\psi}(f)(h)=\prod_{k\in
\psi^{-1}(h)}f(k)$ for every $h\in H$. \end{lem}

These functoriality properties can now be joined to give a connection between different bracketing of iterated wreath products:
\begin{lem}
\label{induction step}  Let $A,B,C$ be finite groups and $A$
abelian. Then there are epimorphisms: $$A\wr (B \wr C)\rightarrow (A\wr
B)\wr C\rightarrow (A\times B)\wr C.$$ \end{lem}
\begin{proof} Let us first construct an
epimorphism $h_*:(A\wr B)\wr C\rightarrow (A\times B)\wr C$. Define $h:A\wr
B\rightarrow A\times B$ by: $$h(f,b)=(\prod_{x\in B}f(x),b),$$ for any
$f:B\rightarrow A,b\in B$. Since $A$ is abelian $h$ is a homomorphism. For
every $a\in A$, let $f_a:B\rightarrow A$ be the map $f_a(b')=0$ for any
$1\not=b'\in B$ and $f_a(e)=a$.   Then clearly $h(f_a,b)=(a,b)$
for any $a\in A,b\in B$ and hence $h$ is onto. By Lemma
\ref{functoriality of first argument}, $h$ induces an epimorphism
$h_*:(A\wr B)\wr C\rightarrow (A\times B)\wr C$. To construct the
epimorphism $A\wr (B \wr C)\rightarrow (A\wr B)\wr C$, we shall use the
associativity of the permutational wreath product (see \cite[Theorem 3.2]{B}).
Using this associativity one has: $$(A\wr B)\wr C= (A\wr_B
B)\wr_C C\cong A\wr_{B\times C} (B\wr_C C).$$ It is now left to construct an
epimorphism: $$ A\wr (B\wr C)=A\wr_{B\wr C} (B\wr C)\rightarrow
A\wr_{B\times C} (B\wr C).$$ By Lemma \ref{construction of G-maps},
there is a $B\wr C$-map $\phi:B\wr C\rightarrow B\times C$ and hence by
Lemma \ref{G-map induces an epimophism} there is an epimorphism
$A\wr_{B\wr C} (B\wr C)\rightarrow A\wr_{B\times C} (B\wr C)$. \end{proof}

Let us iterate Lemma \ref{induction step}. Let $G_1,...,G_n$ be groups. The \it ascending iterated standard wreath product \rm of $G_1,...,G_n$ is
defined as $$(\cdots ((G_1 \wr G_2) \wr G_3)\wr \cdots) \wr G_n,$$ and the  \it descending iterated
standard wreath product \rm of $G_1,...,G_n$ is defined as $$G_1 \wr (G_2 \wr (G_3 \wr \cdots \wr
G_n))\cdots ).$$  These two iterated wreath products are not isomorphic in general, as the standard wreath
product is not associative (as opposed to the ``permutation" wreath product). We shall abbreviate and write $G_1\wr (G_2\wr ... \wr G_n)$ to refer to the descending wreath product and $(G_1 \wr ... \wr G_{r-1})\wr G_r$ to refer to the ascending wreath product.

By iterating the epimorphism in Lemma
\ref{induction step} one obtains:
\begin{cor}\label{decending to ascending} Let $A_1,..,A_r$ be abelian groups. Then $(A_1 \wr ... \wr A_{r-1})\wr A_r$ is an  epimorphic image of  $A_1 \wr ( A_2 \wr ... \wr A_r).
\end{cor}
\begin{proof} By induction on $r$. The cases $r=1,2$ are trivial; assume $r\geq 3$. By the induction hypothesis there is an epimorphism $$\pi_1':A_1 \wr (A_2\wr .... \wr A_{r-1})\rightarrow (A_1\wr ... \wr A_{r-2})\wr A_{r-1}.$$ By Lemma \ref{functoriality of first argument}, $\pi_1'$ induces an epimorphism $\pi_1:(A_1\wr (A_2\wr ... \wr A_{r-1}))\wr A_r \rightarrow (A_1\wr ... \wr A_{r-1})\wr A_r$.
Applying Lemma \ref{induction step} with $A=A_1,B=A_2\wr (A_3\wr ... \wr A_{r-1}), C=A_r$, one obtains an epimorphism:
$$\pi_2: A_1\wr (A_2\wr ... \wr A_r)\rightarrow (A_1\wr (A_2\wr ... \wr A_{r-1}))\wr A_r.$$ Taking the composition $\pi=\pi_1\pi_2$ one obtains an epimorphism $$\pi:A_1\wr (A_2\wr ... \wr A_r)\rightarrow (A_1\wr ... \wr A_{r-1})\wr A_r.$$

\end{proof}
\subsection{Dimension under epimorphisms}
Let us understand how the ``dimension" $\dg$ behaves under the homomorphisms in Lemma \ref{induction step} and Corollary \ref{decending to ascending}. By \cite{KL}, for any finite group $G$ that is not perfect, i.e. $[G,G]\not=G$, where $[G,G]$ denotes the commutator subgroup of $G$, one has $\dg(G)=\dg(G/[G,G])$. According to our definitions, for a perfect group $G$, $\dg(G/[G,G])=\dg(\{1\})=0$, but if $G$ is nontrivial,  $\dg(G)\geq 1$. As nontrivial semiabelian groups are not perfect, this difference will not effect any of the arguments in the sequel.
\begin{defn} Let $G$ be  a finite group and $p$ a prime. Define $\dg_p(G)$ to be the rank of the $p$-Sylow subgroup of $G/[G,G]$, i.e. $\dg_p(G):=\dg((G/[G,G])(p))$.
\end{defn}
Note that if $G$ is not perfect one has $\dg(G)=max_{p}(\dg_p(G))$.

Let $p$ be a prime. An epimorphism $f:G\rightarrow H$ is called $\dg$-preserving (resp. $\dg_p$-preserving) if $\dg(G)=\dg(H)$ (resp. $\dg_p(G)=\dg_p(H)$).

\begin{lem} Let $G$ and $H$ be two finite groups. Then:
$$H\wr G/[H\wr G,H\wr G]\cong H/[H,H]\times G/[G,G].$$
\end{lem}

\begin{proof} Applying Lemmas \ref{functoriality of first argument} and \ref{functoriality with second argument} one obtains an epimorphism $$H\wr G\rightarrow H/[H,H]\wr G/[G,G].$$ By Lemma \ref{induction step} (applied with $C=1$) there is an epimorphism $$H/[H,H]\wr G/[G,G]\rightarrow H/[H,H]\times G/[G,G].$$ Composing these epimorphisms one obtains an epimorphism $$\pi:H\wr G\rightarrow H/[H,H]\times G/[G,G],$$
that sends an element $(f:G\rightarrow H,g)\in H\wr G$ to $$(\prod_{x\in G}f(x)[H,H],g[G,G])\in H/[H,H]\times G/[G,G].$$
The image of $\pi$ is abelian and hence $\kernel(\pi)$ contains $K:=[H\wr G,H\wr G]$.

Let us show $K\supseteq \kernel(\pi)$. Let $(f,g)\in \kernel(\pi)$. Then $g\in [G,G]$ and $\prod_{x\in G}f(x)\in [H,H]$. As $g\in [G,G]$, it suffices to show that the element $f=(f,1)\in H\wr G$ is in $K$. Let $g_1,...,g_n$ be the elements of $G$ and for every $i=1,...,n,$ let $f_i$ be the function for which $f_i(g_i)=f(g_i)$ and $f(g_j)=1$ for every $j\not= i$. One can write $f$ as $\prod_{i=1}^n f_i$. Now for every $i=1,..,n$, the function $f_{1,i}=f_i^{g_i^{-1}}$ satisfies $f_{1,i}(1)=f(g_i)$ and $f_{1,i}(g_j)=1$ for every $j\not=1$. Thus $f_i$ is a product of an element in $[H^{|G|},G]$ and $f_{i,1}$. So,  $f$ is a product of elements in $[H^{|G|},G]$ and $f'=\prod_{i=1}^n f_{1,i}$. But $f'(1)=\prod_{x\in G}f(x)\in [H,H]$ and $f'(g_i)=1$ for every $i\not= 1$ and hence $f'\in [H^{|G|},H^{|G|}]$. Thus, $f\in K$ as required and $K=\kernel{\pi}$.

\end{proof} The following is an immediate conclusion:
\begin{cor} Let $G$ and $H$ be two finite groups. Then $$\dg_p(H\wr G)=\dg_p(H)+\dg_p(G)$$ for any prime $p$.
\end{cor}

So, for groups $A,B,C$ as in Lemma \ref{induction
step}, we have:
$$ \dg_p(A\wr (B \wr C))=\dg_p( (A\times B)\wr C) = \dg_p(A\times B\times C)= \dg_p(A)+\dg_p(B)+\dg_p(C) $$ for every $p$.
In particular, the epimorphisms in Lemma \ref{induction step}  are $\dg$-preserving.

The same observation holds for Corollary \ref{decending to ascending}, so one has:
\begin{lem} Let $A_1,...,A_r$ be finite abelian groups. Then
$$ \dg_p(A_1\wr (A_2\wr ...\wr A_r))=\dg_p((A_1\wr ... \wr A_{r-1})\wr A_r)=\dg_p(A_1\times... \times A_r)$$ are all $\sum_{i=1}^r\dg_p(A_i)$ for any prime $p$.
\end{lem}
For cyclic groups $A_1,...,A_r$, $\dg_p(A_1 \wr (A_2 \wr ... \wr A_r))$ is simply the number of cyclic groups among $A_1,..,A_r$ whose $p$-part is non-trivial. Thus:
\begin{cor}\label{d of iterated cyclic is max appearence of p} Let $C_1,...,C_r$ be finite cyclic groups and $G=C_1\wr (C_2 \wr ...\wr C_r)$. Then
$\dg(G)=max_{p||{G}|}\dg(C_1(p)\wr (C_2(p) \wr ... \wr C_r(p)))$.
\end{cor}

Let us apply Lemma \ref{induction step} in order to connect between descending iterated wreath products of abelian and cyclic groups:
\begin{prop}\label{connection between abelian and cyclic} Let $A_1,...,A_r$ be finite abelian groups and let $A_i$ have invariant factors $C_{i,j}$ for $j=1,..,l_i$, i.e. $A_i=\prod_{j=1}^{l_i}C_{i,j}$ and $|C_{i,j}|||C_{i,j+1}|$ for any $i=1,...,r$ and $j=1,...,l_i-1$.
Then there is an epimorphism from the descending iterated wreath product $\widetilde{G}:=\wr_{i=1}^r \wr_{j=1}^{l_i} C_{i,j}$ {\rm(here the groups $C_{i,j}$ are ordered lexicographically: $C_{1,1}, C_{1,2},...,C_{1,l_1},C_{2,1},...,C_{r,l_r}$)} to $G:= A_1\wr (A_2 \wr ...\wr A_r)$.
\end{prop}
\begin{proof}

Let us assume $A_1\not=\{0\}$ (otherwise $A_1$ can be simply omitted). Let us prove the assertion by induction on  $\sum_{i=1}^rl_i$. Let $G_2=A_2\wr (A_3 \wr ... \wr A_k)$. Write $A_1=C_{1,1}\times A_1'$. By Lemma \ref{induction step}, there is an  epimorphism $\pi_1:C_{1,1} \wr (A_1'\wr G_2)\rightarrow (C_{1,1}\times A_1')\wr G_2=A_1\wr G_2=G$.
By applying the induction hypothesis to $A_1',A_2,...,A_r$, there is an epimorphism $\pi_2'$ from the  descending iterated wreath product $\widetilde{G}_2=\wr_{j=2}^{l_1} C_{1,j} \wr (\wr_{i=2}^r \wr_{j=1}^{l_i} C_{i,j})$ to $A_1'\wr G_2$.
By Lemma \ref{functoriality with second argument}, $\pi_2'$ induces an epimorphism $\pi_2:C_{1,1}\wr \widetilde{G}_2\rightarrow C_{1,1} \wr (A_1'\wr G_2)$. Taking the composition $\pi=\pi_2\pi_1$, we obtain the required epimorphism:
$ \pi: \widetilde{G}=C_{1,1}\wr \widetilde{G}_2\rightarrow G. $
\end{proof}
\begin{rem} Note that:
$$ \dg_p(\widetilde{G})=\sum_{i=1}^r\sum_{j=1}^{l_i} \dg_p(C_{i,j})=\sum_{i=1}^r \dg_p(A_i)=\dg_p(G)$$
for every $p$ and hence $\pi$  is $\dg$-preserving.
\end{rem}
Therefore, showing $G$ is a $\dg$-preserving epimorphic image of an iterated wreath product of abelian groups is equivalent to showing $G$ is a $\dg$-preserving epimorphic image of an iterated wreath product of finite cyclic groups.

\section{Wreath length}

The following lemma is essential for the definition of wreath length:
\begin{lem} Let $G$ be a finite semiabelian group. Then $G$ is a homomorphic
image of a descending iterated wreath product  of finite cyclic
groups, i.e. there are finite cyclic groups $C_1,...,C_r$ and an
epimorphism $C_1\wr (C_2\wr ... \wr C_r)\rightarrow G.$
\end{lem}
\begin{proof} By Proposition \ref{connection between abelian and cyclic} it suffices to show $G$ is an epimorphic image of a descending iterated wreath product of finite abelian groups. We shall prove this claim by induction on $|G|$. The case $G=\{1\}$ is trivial.  By \cite{Den}, $G=A_1H$ with $A_1$ an abelian normal subgroup and $H$ a proper semiabelian subgroup of $G$. First, there is an epimorphism $\pi_1:A_1\wr H \rightarrow A_1H=G$.  By induction there are abelian groups $A_2,..,A_r$ and an epimorphism $\pi_2':A_2 \wr (A_3 \wr ... \wr A_r)\rightarrow H$. By Lemma \ref{functoriality of first argument}, $\pi_2'$ can be extended to an epimorphism $\pi_2:A_1 \wr (A_2\wr ... \wr A_r) \rightarrow A_1 \wr H$. So, by taking the composition $\pi=\pi_1\pi_2$ one obtains the required epimorphism $\pi:A_1 \wr (A_2 \wr ... \wr A_r) \rightarrow G$.
 \end{proof}
We can now define:
\begin{defn} Let $G$ be a finite semiabelian group.  Define the {\it wreath length} $\wl(G)$ of $G$ to be the smallest positive integer $r$ such that there are finite cyclic groups $C_1,...,C_r$ and an
epimorphism $C_1\wr (C_2 \wr ... \wr C_r) \rightarrow G$.
\end{defn}
Let $\widetilde{G}=C_1\wr (C_2 \wr ... \wr C_r)$ and $\pi:\widetilde{G}\rightarrow G$ an epimorphism. Then by Corollary \ref{d of iterated cyclic is max appearence of p}: $$ \dg(G)\leq \dg(\widetilde{G})\leq r.$$ In particular $\dg(G)\leq \wl(G)$.
\begin{prop}\label{wreath len of wreath prd} Let $C_1,...,C_r$ be nontrivial finite cyclic groups.  Then $\wl(C_1\wr (C_2 \wr ...\wr C_r))=r$.
\end{prop}
Let $\dl(G)$ denote the derived length of a (finite) solvable group $G$, i.e. the smallest positive integer $n$ such that the $n$th higher commutator subgroup of $G$ ($n$th element in the derived series $G=G^{(0)}\geq G^{(1)}=[G,G]\geq \cdots \geq G^{(i)}=[G^{(i-1)},G^{(i-1)}]\geq \cdots$) is trivial.  In order to prove this proposition we will use the following lemma:
\begin{lem}\label{derived length lemma}   Let $C_1,...,C_r$ be nontrivial finite cyclic groups.  Then $\dl(C_1\wr (C_2 \wr ...\wr C_r))=r$.
\end{lem}
\begin{proof}
It is easy (by induction) to see that $\dl(C_1\wr (C_2 \wr ...\wr C_r))\leq r$.  We turn to the reverse inequality.  By Corollary 2.11, it suffices to prove it for the ascending iterated wreath product
$G=(C_1 \wr ... \wr C_{r-1})\wr C_r$. We prove this by induction on $r$.  The case $r=1$ is trivial.  Assume $r\geq 1$. Write $G_1:=(C_1 \wr ... \wr C_{r-2}) \wr C_{r-1}$ so that $G=G_1\wr C_r$. By induction hypothesis, $\dl(G_1)=r-1.$   View $G$ as the semidirect product $G_1^r\rtimes C_r$.  For any $g\in G_1$, the element $t_g:=(g,g^{-1},1,1,...,1)\in G_1^r$ lies in $[G_1^r,C_r]$ and hence in $[G_1^r,C_r]\leq G' \leq G_1^r$.  Let $H=\{t_g|g\in G_1 \}$.  The projection map $G_1^r\rightarrow G_1$ onto the first copy of $G_1$ in $G_1^r$ maps $H$ onto $G_1$.  Since $H\leq G'$, the projection map also maps $G'$ onto $G_1$.  Now $\dl(G_1)=r-1$ by the induction hypothesis.  It follows that $\dl(G')\geq r-1$, whence $\dl(G)\geq r$.
\end{proof}
To prove the proposition, we first observe that  $\wl(C_1\wr (C_2 \wr ...\wr C_r))\leq r$ by definition.  If
$C_1\wr (C_2 \wr ...\wr C_r)$ were a homomorphic image of a shorter descending iterated wreath product
$C_1'\wr (C_2' \wr ...\wr C_s')$, then by Lemma \ref{derived length lemma}, $s=\dl(C_1'\wr (C_2' \wr ...\wr C_s'))\geq \dl(C_1\wr (C_2 \wr ...\wr C_r))=r>s$, contradiction. \qed

Combining Proposition \ref{wreath len of wreath prd} with Corollary \ref{d of iterated cyclic is max appearence of p} we have:
\begin{cor}\label{descriptio of wl=d for iterated} Let $C_1,...,C_r$ be finite cyclic groups and $G=C_1\wr (C_2 \wr ...\wr C_r)$. Then $\wl(G)=\dg(G)$ if and only if there is a prime $p$ for which $p||C_1|,...,|C_r|$.
\end{cor}

We shall now see that all examples of groups $G$ with $\wl(G)=\dg(G)$ arise from Corollary \ref{descriptio of wl=d for iterated}:
\begin{prop}\label{out char of wl} Let $G$ be a finite semiabelian group. Then $\wl(G)=\dg(G)$ if and only if there is a prime $p$, finite cyclic groups $C_1,...,C_r$ for which $p||C_i|$, $i=1,...,r$,  and a $\dg$-preserving epimorphism $\pi:C_1 \wr (C_2 \wr ... \wr C_r)\rightarrow G$.
\end{prop}
\begin{proof} Let $d=\dg(G)$. The equality $d=\wl(G)$ holds if and only if there are finite cyclic groups $C_1, C_2,...,C_d$ and an epimorphism $\pi:\widetilde{G}=C_1 \wr (C_2 \wr ... \wr C_d)\rightarrow G$.
Assume the latter holds. Clearly  $d\leq \dg(\widetilde{G})$ but by Corollary \ref{d of iterated cyclic is max appearence of p} applied to $\widetilde{G}$ we also have $\dg(\widetilde{G})\leq d$. It follows that $\pi$ is $\dg$-preserving. Since $\dg(G)=max_p(\dg_p(G))$, there is a prime $p$ for which $d=\dg_p(G)$ and hence $\dg_p(\widetilde{G})=d$. Thus, $p||C_i|$ for all $i=1,...,r$.

Let us prove the converse. Assume there is a prime $p$, finite cyclic groups $C_1,...,C_r$ for which $p||C_i|$, $i=1,...,r$,  and a $\dg$-preserving epimorphism $\pi:\widetilde{G}:=C_1 \wr (C_2 \wr ... \wr C_r)\rightarrow G$. Since $p||C_i|$, it follows that $\dg_p(\widetilde{G})=r$. As $\dg_p(\widetilde{G})\leq \dg(\widetilde{G})\leq r$, it follows that $\dg(G)=\dg(\widetilde{G})=r$. In particular $\wl(G)\leq r=\dg(G)$ and hence $\wl(G)=\dg(G)$.

\end{proof}

\begin{rem}\label{cyclic dec of p-groups}
Let $G$ be a semiabelian $p$-group.
By \cite[Corollary 2.15]{N}, $G$ is a $\dg$-preserving image of an iterated wreath product of abelian
subgroups of $G$ (following the proof one can observe that the abelian groups were
actually subgroups of $G$). So, by Proposition \ref{connection between abelian and cyclic},
$G$ is a $\dg$-preserving epimorphic image of $\widetilde{G}:=C_1 \wr (C_2 \wr ... \wr C_k)$
for cyclic subgroups $C_1,...,C_k$ of $G$. By applying Proposition \ref{out char of wl} one obtains $\wl(G)=\dg(G)$.
\end{rem}
\begin{rem} Throughout the proof of \cite[Corollary 2.15]{N} one can use the minimality assumption posed on the decompositions to show directly that the abelian groups $A_1,...,A_r$, for which there is a $\dg$-preserving epimorphism $A_1 \wr (A_2 \wr ... \wr A_r)\rightarrow G$, can be actually chosen to be cyclic.
\end{rem}

We shall generalize Remark \ref{cyclic dec of p-groups} to nilpotent groups:

\begin{prop}\label{wl=d for nilp} Let $G$ be a finite nilpotent semiabelian group. Then $\wl(G)=\dg(G)$
\end{prop}
\begin{proof} Let $d=\dg(G)$. Let $p_1,...,p_k$ be the primes dividing $|G|$ and let $P_i$ be the $p_i$-Sylow subgroup of $G$ for every $i=1,...,k$. So, $G\cong \prod_{i=1}^k P_i$. By Remark \ref{cyclic dec of p-groups}, there are cyclic $p_i$-groups $C_{i,1},...,C_{i,r_i}$ and a
$\dg$-preserving epimorphism $\pi_i:C_{i,1} \wr (C_{i,2} \wr ... \wr C_{i,r_i})\rightarrow P_i$ for every $i=1,..,k$.
In particular for any $i=1,...,k$, $r_i= \dg(P_i)=\dg_p(G)\leq d$.  For any $i=1,..,k$ and any $d\geq j>r_i$, set $C_{i,j}=\{1\}$. For any $j=1,...,d$
define $C_j=\prod_{i=1}^kC_{i,j}$.

We claim $G$ is an epimorphic image of $\widetilde{G}=C_1 \wr (C_2 \wr ... \wr C_{d})$. To prove this claim it suffices
to show every $P_i$ is an epimorphic image of $\widetilde{G}$ for every $i=1,..,k$. As $C_{i,j}$ is an epimorphic image of $C_j$ for every
$j=1,...,d$ and every $i=1,..,k$, one can apply Lemmas \ref{functoriality of first argument}  and \ref{functoriality with second argument} iteratively to obtain an epimorphism $\pi_i':\widetilde{G}\rightarrow C_{i,1} \wr (C_{i,2} \wr ... \wr C_{i,r})$ for every $i=1,...,k$. Taking the
composition $\pi_i'\pi_i$ gives the required epimorphism and proves the claim.  As $G$ is an epimorphic image of an iterated wreath product of $\dg(G)$ cyclic groups
one has $\wl(G)\leq \dg(G)$ and hence $\wl(G)=\dg(G)$.

\end{proof}

\begin{exam} Let $G=D_n=\langle \sigma,\tau | \sigma^2=1,\tau^n=1, \sigma\tau\sigma=\tau^{-1} \rangle$ for $n\geq 3$. Since $G$ is an epimorphic image of $\langle\tau\rangle\wr \langle\sigma\rangle$ and $G$ is not abelian we have $\wl(G)=2$.
On the other hand $\dg(G)=\dg(G/[G,G])$ is $1$ if $n$ is odd and $2$ if $n$ is even. So, $G=D_3=S_3$ is the minimal example for which $\wl(G)\not=\dg(G)$.
\end{exam}

\section{a ramification bound for semiabelian groups} In
this section we prove:

\begin{thm}\label{main Theorem} Let $G$ be a
finite semiabelian group.  Then there exists a tamely ramified extension $K/\mathbb{Q}$ with $G(K/\mathbb{Q})\cong G$
in which at most $\wl(G)$ primes ramify.  \end{thm}

The
proof relies on the splitting Lemma from
\cite{KS}: Let $\ell$ be a rational prime, $K$ a number field and
$\frak{p}$
a prime of $K$ that is prime to $\ell$.
Let $I_{K,\frak{p}}$ denote the group of fractional ideals prime to
$\frak{p}$, $P_{K,\frak{p}}$ the subgroup of principal ideals that are prime
to $\frak{p}$ and let  $P_{K,\frak{p},1}$ be the subgroup of principal ideals
$(\alpha)$ with $\alpha\equiv 1$ (mod $\frak{p}$). Let $\overline
P_{\frak{p}}$ denote $P_{K,\frak{p}}/P_{K,\frak{p},1}$. The ray class group
$Cl_{K,\frak{p}}$ is defined to be $I_{K,\frak{p}}/P_{K,\frak{p},1}$. Now, as
$I_{K,\frak{p}}/P_{K,\frak{p}}\cong Cl_{K}$, one has the following short exact
sequence: \begin{equation} \label{ray class field sequence}
1 \longrightarrow \overline
P_{\frak{p}}^{(\ell)} \longrightarrow Cl_{K,\frak{p}}^{(\ell)} \longrightarrow
Cl_{K}^{(\ell)} \longrightarrow 1, \end{equation} where $A^{(\ell)}$
denotes the $\ell$-primary component of an abelian group $A$. Let us
describe a sufficient condition for the splitting of (\ref{ray class
field sequence}). Let $\frak{a}_1,...,\frak{a}_r \in I_{K,\frak{p}}$,
$\tilde\frak{a}_1,...,\tilde\frak{a}_r$ their classes in $Cl_{K,\frak{p}}^{(\ell)}$
with images $\overline{\frak{a}}_1,...,\overline{\frak{a}}_r$ in
$Cl_K^{(\ell)}$, so that
$Cl_{K}^{(\ell)}=\langle\overline{\frak{a}}_1\rangle\times
\langle\overline{\frak{a}}_2\rangle\times...\times \langle
\overline{\frak{a}}_r\rangle$.  Let $\ell^{m_i}:=|\langle \overline
\frak{a}_i\rangle |$ and let $a_i\in K$ satisfy
$\frak{a}_i^{\ell^{m_i}}=(a_i)$, for $i=1,...,r$.
\begin{lem}\label{useful part of the splitting
lemma}\emph{(Kisilevsky-Sonn \cite{KS2})} Let $\frak{p}$ be a prime of
$K$ and let $K'=K(\sqrt[\ell^{m_i}]{a_i}|i=1,...,r)$. If $\frak{p}$
splits completely in $K'$ then the sequence (\ref{ray class field
sequence}) splits. \end{lem} The splitting  of (\ref{ray class field
sequence}) was used in \cite{KS} to construct cyclic ramified
extensions at one prime only. Let $m=\max\{1,m_1,...m_r\}$.  Let
$U_K$ denote the units in $\mathcal{O}_K$. \begin{lem}\label{cor-existence
of totally ramified extension}\emph{(Kisilevsky-Sonn \cite{KS})} Let
$K''=K(\mu_{\ell^m},\sqrt[\ell^m]{\xi},\sqrt[\ell^{m_i}]{a_i}|\
\xi\in U_K, i=1,...,r)$ and $\frak{p}$ a prime of $K$ which splits
completely in $K''$. Then there is a cyclic $\ell^m$-extension of
$K$ that is totally ramified at $\frak{p}$ and is not ramified at any
other prime of $K$. \end{lem}

\begin{cor}\label{K'''} Let $K$ be a number field, $n$ a positive integer.  Then there exists a finite extension $K'''$ of $K$ such that if $\frak{p}$ is any prime of $K$ that splits completely in $K'''$, then there exists a cyclic extension $L/K$ of degree $n$ in which $\frak{p}$ is totally ramified and $\frak{p}$ is the only prime of $K$ that ramifies in $L$.
\end{cor}
\begin{proof} Let $n=\prod_{\ell}\ell^{m(\ell)}$ be the decomposition of $n$ into primes.  Let $K'''$ be the composite of the fields $K''=K''(\ell)$ in Lemma \ref{cor-existence of totally ramified extension} ($m=m(\ell))$.  Let $L(\ell)$ be the cyclic extension of degree $\ell^{m(\ell)}$ yielded by Lemma \ref{cor-existence of totally ramified extension}.  The composite $L=\prod L(\ell)$ has the desired property.
\end{proof}

\begin{proof}(Theorem \ref{main Theorem})  \rm By definition, $G$ is a homomorphic image of a descending iterated wreath product of cyclic groups $C_1 \wr (C_2 \wr \cdots \wr C_r)$, $r=\wl(G)$.  Without loss of generality $G\cong C_1 \wr (C_2 \wr \cdots \wr C_r)$ is itself a  descending iterated wreath product of cyclic groups.  Proceed by induction on $r$.  For $r=1$, $G$ is cyclic of order say $N$. If $p$ is a rational prime $\equiv 1$
(mod $N$), then the field of $p$th roots of unity $\mathbb{Q}(\mu_p)$
contains a subfield $L$ cyclic over $\mathbb{Q}$ with Galois group $G$ and
exactly one ramified prime, namely $p$.  Thus the theorem holds for
$r=1$.

Assume $r>1$ and the theorem holds for $r-1$.  Let $K_1/\mathbb{Q}$ be a tamely ramified Galois extension with $G(K_1/\mathbb{Q})\cong G_1$, where $G_1$ is the descending iterated wreath product $C_2 \wr (C_3 \wr \cdots \wr C_r)$, such that the ramified primes in $K_1$ are a subset of $\{p_2,...,p_r\}$.  By Corollary \ref{K'''}, there exists a prime $p=p_1$ not dividing the order of $G$ which splits completely in $K_1'''$, the field supplied for $K_1$ by Corollary \ref{K'''}, and let $\frak{p}=\frak{p}_1$ be a prime of $K_1$ dividing $p$.  By Corollary \ref{K'''}, there exists a cyclic extension $L/K_1$ with $G(L/K_1)\cong C_1$ in which $\frak{p}$ is totally ramified and in which $\frak{p}$ is the only prime of $K_1$ which ramifies in $L$.

Now $\frak{p}$ has $|G_1|$ distinct conjugates $\{\sigma(\frak{p})|\sigma \in
G(K_1/\mathbb{Q})\}$ over $K_1$.  For each $\sigma \in G(K_1/\mathbb{Q})$, the conjugate extension
$\sigma(L)/K_1$ is well-defined, since $K_1/\mathbb{Q}$ is Galois.  Let $M$ be the
composite of the   $\sigma(L)$, $\sigma \in G(K_1/\mathbb{Q})$.  For each $\sigma$,
$\sigma(L)/K_1$ is cyclic of degree $|C_1|$, ramified only at $\sigma(\frak{p})$, and
$\sigma(\frak{p})$ is totally ramified in $\sigma(L)/K_1$.  It now follows (see e.g. \cite[Lemma 1]{KS}) that the fields $\{\sigma(L)|\sigma \in G(K_1/\mathbb{Q})\}$ are linearly disjoint over
$K_1$, hence $G(M/\mathbb{Q})\cong C_1 \wr  G_1 \cong G$. Since the only primes of
$K_1$ ramified in $M$ are $\{\sigma(\frak{p})|\sigma \in G(K_1/\mathbb{Q})\}$, the only rational
primes ramified in $M$ are $p_1,p_2,...,p_n$. \end{proof}

\begin{cor} The minimal ramification problem has a positive solution for all finite semiabelian groups $G$ for which $\wl(G)=\dg(G)$.  Precisely, any finite semiabelian group $G$ for which $\wl(G)=\dg(G)$ can be realized tamely as a Galois group over the rational numbers with exactly $\dg(G)$ ramified primes. \end{cor}

By Proposition \ref{wl=d for nilp}, we have

\begin{cor} The minimal ramification problem has a positive solution for all finite nilpotent semiabelian groups.  \end{cor}

\section{Arithmetic consequences}

\bigskip
In this section we examine some arithmetic consequences of a
positive solution to the minimal ramification problem.
Specifically, given a group $G,$ the existence of infinitely many
minimally tamely ramified $G$-extensions $K/\mathbb{Q}$ is re-interpreted
in some cases in terms of some open problems in algebraic
number theory. We will be most interested in the case $\dg(G)=1.$

\begin{prop}{\label{prop:classnumber divisibility}
Let $q$ and $\ell$ be distinct primes. Let $K/\mathbb{Q}$ be a cyclic extension of degree  $n:=[K:\mathbb{Q}]\geq 2$ with $(n,q\ell)=1$. Suppose that $K/\mathbb{Q}$ is totally and tamely ramified  at a unique prime $\mathfrak l$ dividing $\ell$.  Then $q$ divides the class number $h_K$ of $K$ if and only if
there exists an extension $L/K$ satisfying the following:
\item{i).} $L/\mathbb{Q}$ is a Galois extension with {\bf non-abelian} Galois group $G=G(L/\mathbb{Q})$.
\item{ii).} The degree $[L:K]=q^s$ is a power of $q.$
\item{iii).} $L/\mathbb{Q}$ is (tamely) ramified only at primes over $\ell.$}
\end{prop}

\begin{proof}
 First suppose that $q$ divides $h_K.$  Let $K_0$ be the $q$-Hilbert class field
of $K,$ {\it i.e.\/} $K_0/K$ is the maximal unramified abelian $q$-extension of $K.$
Then $K_0/\mathbb{Q}$ is a Galois extension with Galois group $G:=G(K_0/\mathbb{Q}),$
and $H:=G(K_0/K)\simeq (C_K)_q \neq 0$, the $q$-part
of the ideal class group of $K.$ Then $[G,G]$ is contained in $H.$
If $[G,G] \subsetneq H,$ then the fixed field of $[G,G]$ would be an abelian extension of $\mathbb{Q}$ which
contains an unramified $q$-extension of $\mathbb{Q}$ which is impossible. Hence $[G,G]=H\neq 0$ and so $G$ is
a non-abelian group, and $L=K_0$ satisfies {\it i),ii)}, and {\it iii)} of the statement.

Conversely suppose that there is an extension $L/K$ satisfying {\it i),ii)}, and {\it iii)} of the statement.
Since $H=G(L/K)$ is a $q$-group, there is a sequence of normal subgroups
$H=H_0\supset H_1\supset H_2 \cdots \supset H_s=0$ with $H_i/H_{i+1}$ a cyclic
group of order $q.$ Let $L_i$ denote the fixed field of $H_i$ so that
$K=L_0\subset \cdots\subset L_s=L.$ Let $m$ be the largest index such that $L_m/\mathbb{Q}$ is totally ramified
(necessarily at $\ell$). If $m=s,$ then $L/\mathbb{Q}$ is totally and tamely ramified at $\ell$ and so the inertia
group $T(\mathfrak L/(\ell))=G,$ where in this case $\mathfrak L$ is the unique prime of $L$ dividing $\ell.$
Since $L/\mathbb{Q}$ is tamely ramified it follows that  $T(\mathfrak L/(\ell))$ is cyclic, but this contradicts
the hypothesis that $G$ is non-abelian. Therefore it follows that $m<s,$ and so $L_{m+1}/L_m$ is unramified
and therefore $q$ must divide the class number $h_{L_m}.$ Then a result of Iwasawa \cite{I} implies that $q$
divides all of the class numbers $h_{L_{m-1}}, \cdots,h_{L_0}=h_K.$
\end{proof}

We now apply this to the case that $G\not=\{1\}$ is a quotient of the regular wreath product $C_q \wr C_p$
where $p$ and $q$ are distinct primes. Then $\dg(G)=1.$

The existence of infinitely many minimally tamely ramified $G$-extensions $L/\mathbb{Q}$ would by
Proposition \ref{prop:classnumber divisibility} imply the existence of infinitely many
cyclic extensions $K/\mathbb{Q}$ of degree $[K:\mathbb{Q}]=p$ ramified at a unique prime $\ell \neq p,q$
for which $q$ divides the class number $h_K.$
(If there were only finitely many distinct such cyclic extensions $K/\mathbb{Q},$
then the number of ramified primes $\ell$ would be bounded, and there would be an absolute upper bound
on the possible discriminants of the distinct fields $L/\mathbb{Q}.$ By Hermite's theorem, this would mean that
the number of such $G$-extensions $L/\mathbb{Q}$ would be bounded).

The question of whether there is an infinite number of cyclic degree $p$ extensions (or even one) of $\mathbb{Q}$ whose
class number is divisible by $q$ is in general open at this time.

For $p=2,$ it is known that
there are infinitely many quadratic fields (see Ankeny, Chowla \cite{AC}), with class numbers divisible
by $q,$ but it is not known that this occurs for quadratic fields with prime discriminant.

This latter statement is also a consequence of Schinzel's hypothesis as is shown by Plans in \cite{P}.
There is also some numerical evidence that the heuristic of Cohen-Lenstra should be statistically independent of
the primality of the discriminant (see Jacobson, Lukes, Williams \cite{JLW} or te Riele, Williams \cite{RW}).
If this were true, then one would expect that there is a positive density of primes $\ell$
for which the cyclic extension of degree $p$ and conductor $\ell$ would have class number divisible by $q.$

For $p=3$ it has been proved by Bhargava \cite{B} that there are infinitely many cubic fields $K/\mathbb{Q}$
for which $2$ divides their class numbers. That there are infinitely many cyclic cubics with prime
squared discriminants whose class numbers are even (or more generally divisible by some fixed prime $q$)
seems out of reach at this time.

In our view, there is a significant arithmetic interest in solving the minimal ramification problem for other groups (see also \cite{H}, \cite{JR}, \cite{R}).

redpajama_set_name: RedPajamaArXiv