Question: In the Pokemon School, you can create a group and other players can join. I created and some friends of mine joined. The NPC says something about syncing events.

What does being in a group do? What kind of things does it sync?

Answer: 
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> Joining a group is a feature introduced in Generation IV. Players in the same group encounter the same swarming Pokémon, weather conditions, changing Pokémon in the Great Marsh, Feebas location, and other things each day. Group members can compare records on the third floor of Jubilife TV.
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Source: [Bulbapedia](http://bulbapedia.bulbagarden.net/wiki/Groups)
I always created a group with my friend when we wanted to enter the Doubles Battle Tower together. I assume that "event" in this context refers to any multiplayer event that can be done with a friend in HGSS. They use groups for all of them so that it remains consistent.