text: # Thread: How to prove a monoid is a group.
  
  1. ## How to prove a monoid is a group.
  
  Let c be a fixed positive integer, and let * denote the binary operation on the set Z of integers defined be the formula
  
  $x * y = xy + c(x+y) + c^2 - c$
  
  for all integers x, y, and z.
  
  Is (Z, *) a group?
  
  I already proved that it's a monoid. Do I have to do this step by induction? Could someone guide me towards a solution?
  
  2. Originally Posted by pikminman
  Let c be a fixed positive integer, and let * denote the binary operation on the set Z of integers defined be the formula
  
  $x * y = xy + c(x+y) + c^2 - c$
  
  for all integers x, y, and z.
  
  Is (Z, *) a group?
  
  I already proved that it's a monoid. Do I have to do this step by induction? Could someone guide me towards a solution?
  A group is a monoid where every element has an inverse, $a*a' = e$ where $e$ is your identity and $a$ your element. So, you just plug in you $a$ and $a'$ and see what $a'$ must be for $a*a' = e$.
  
  In these questions always look out for possible illegal steps, such as dividing by zero. Often most elements will have an inverse, but some wont, and the ones which wont will be the ones where zero is in the botton of your inverse...
explanation: The text extract is a discussion thread from a forum where a user asks how to prove that a monoid is a group. The user provides a specific binary operation and asks whether the set of integers with this operation forms a group. Another user responds by explaining that to prove it is a group, the user needs to show that every element has an inverse. The response also mentions to be cautious of potential illegal steps, such as division by zero.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Thread: How to prove a monoid is a group.
  
  1. ## How to prove a monoid is a group.
  
  Let c be a fixed positive integer, and let * denote the binary operation on the set Z of integers defined be the formula
  
  $x * y = xy + c(x+y) + c^2 - c$
  
  for all integers x, y, and z.
  
  Is (Z, *) a group?
  
  I already proved that it's a monoid. Do I have to do this step by induction? Could someone guide me towards a solution?
  
  2. Originally Posted by pikminman
  Let c be a fixed positive integer, and let * denote the binary operation on the set Z of integers defined be the formula
  
  $x * y = xy + c(x+y) + c^2 - c$
  
  for all integers x, y, and z.
  
  Is (Z, *) a group?
  
  I already proved that it's a monoid. Do I have to do this step by induction? Could someone guide me towards a solution?
  A group is a monoid where every element has an inverse, $a*a' = e$ where $e$ is your identity and $a$ your element. So, you just plug in you $a$ and $a'$ and see what $a'$ must be for $a*a' = e$.
  
  In these questions always look out for possible illegal steps, such as dividing by zero. Often most elements will have an inverse, but some wont, and the ones which wont will be the ones where zero is in the botton of your inverse...
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract is a discussion thread from a forum where a user asks how to prove that a monoid is a group. The user provides a specific binary operation and asks whether the set of integers with this operation forms a group. Another user responds by explaining that to prove it is a group, the user needs to show that every element has an inverse. The response also mentions to be cautious of potential illegal steps, such as division by zero.
  
  Label: question-and-answer