1
vote
1answer
217 views

Bounds for conjugacy classes of subgroups

My question is rather general and the reason for this is that I am primarily interested in finding sources where I can read more about this type of problems. So here goes: To begin with, I want to ...
27
votes
3answers
2k views

Names of finite groups

Question: If you have a finite group, how do you name it? If, for whatever reason, you have to list all subgroups of $GL_2({\mathbb F}_5)$ up to isomorphism in a paper, you are likely to write ...
3
votes
2answers
669 views

A question about finite groups.

Let $k$ be a positive integer. Is it true that any finite group $H$ of cardinal $4k+2$ whose center contains an element $h$ of order $2$ is isomorphic to the direct product ...
34
votes
5answers
2k views

Heuristic argument that finite simple groups _ought_ to be “classifiable”?

Obviously there exists a list of the finite simple groups, but why should it be a nice list, one that you can write down? Solomon's AMS article goes some way toward a historical / technical ...
33
votes
5answers
3k views

Why are the sporadic simple groups HUGE?

I'm merely a grad student right now, but I don't think an exploration of the sporadic groups is standard fare for graduate algebra, so I'd like to ask the experts on MO. I did a little reading on them ...
31
votes
19answers
5k views

Fun applications of representations of finite groups

Are there some fun applications of the theory of representations of finite groups? I would like to have some examples that could be explained to a student who knows what is a finite group but does not ...
24
votes
11answers
6k views

Lecture notes on representations of finite groups

Next term I am supposed to teach a course on representation of finite groups. This is a third year course for undegrads. I was thinking to use the book of Grodon James and Martin Liebeck ...
38
votes
10answers
2k views

Why are characters so well-behaved?

Last year I attended a first course in the representation theory of finite groups, where everything was over C. I was struck, and somewhat puzzled, by the inexplicable perfection of characters as a ...