Question

What's your favourite group?

Answer 1

The finite simple group of order 1004913. It doesn't exist, but it's really annoying to show that

Answer 2

SO(3) is my favorite. It has so many weird properties. It pretends to be a 3-sphere, but it isn’t. It’s a subset of projective space. And yet it’s so useful in engineering. really the first physical non-Euclidean manifold we’re all introduced too, yet it isn’t an n-sphere, it’s non-commutative, non-constant curvature, yet it has so many nice properties like compactness, connectedness, it’s a Lie group, and clearly it has so many physical interpretations. Not to mention it’s completely represented by the space of orthogonal matrices with unit determinant.

Answer 3

S^1 (circle group)

Answer 4

Free groups! I've heard free objects in general are pretty cool.

Answer 5

Absolute Galois group of Q. (related: My first thought was “Z_p doesn’t form a group under multiplication”. Then I remembered I’m a number theorist. Don’t forget to exclude 0, btw.)

Answer 6

SL2R

Answer 7

Linkin park

Answer 8

The Klein four group is the smallest group that’s not trivial or cyclic which makes it interesting imo. Same goes for S_3

Answer 9

I like GL(n,R) cause i love linear algebra and you can both analyze this group (and it's subgroups) using algebra and using topology

Answer 10

Z/2Z because orientation sucks (e.g. see the construction of Morse homology).