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I would like to know about your favorite group(s). Since groups do appear everywhere in mathematics and there are plenty of them, which ones have drawn your attention the most or surprised you? Please not just name the group, but also provide some facts about it why you find this one particularly interesting.

I'll start by mentioning Grigorchuks group. Because is was the first group i encountered, wich is finitely generated but not finitely presented. Also it was the first group discovered with intermediate growth. It has a lot of "strange" properties like:

  • it's infinite but residually finite

  • it's amenable but not elementary amenable

  • every proper quotient group is finite

  • every maximal subgroup has finite index

Also Grigorchuks group acts as a key-counterexample in infinite group theory. My professor once told me: "If you have a conjecture about infinite groups, try it one Grigorchuks group. If it holds, it might be worth trying to prove it."

edit: flagged for community wiki.

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closed as primarily opinion-based by Lost1, user127.0.0.1, Bruno Joyal, Cameron Buie, arjafi Jan 30 '14 at 16:00

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise.If this question can be reworded to fit the rules in the help center, please edit the question.

The monster group. Its a true monstrosity. – Your Ad Here Jan 30 '14 at 14:54
I think this is a nice question, or rather the answers could be nice. – Martin Brandenburg Jan 30 '14 at 14:59
For much more trivial reasons "my group" is the $PSL(2,7)\cong GL(3,2)$, aka. the smallest simple group if one doesn't count cyclic and alternating groups. This guy crept up everywhere in my first university semester ... – Hagen von Eitzen Jan 30 '14 at 16:22
would it be more appropriate to ask for examples of groups with "strange" (in the sense of counter-intuitive) properties? like grigorchuks group. i think Dietrich Burdes awnser would also count for that, since it's a really small group wich inherits alot of structual implications. i'd really like to know wich groups are interesting to others and why. – quis23 Jan 30 '14 at 16:43
I want to know how Grigorchuks group was the first group you came across? – hmmmm Jan 30 '14 at 20:32

What about the smallest non-trivial group $\mathbb{Z}/2\mathbb{Z}$ ? See the discussion Fantastic properties of $\mathbb{Z}/2\mathbb{Z}$ for many convincing arguments.

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