1
vote
0answers
41 views

Relation between $|H \lor K|$ , $|H|$ and $|K|$

Let $H$ and $K$ be subgroups of a finite group , then we know that the subgroup generated by $H \cup K$ i.e. $H \lor K$ is the smallest subgroup containing both $H$ and $K$ , then how can we relate ...
5
votes
1answer
176 views

Why do we care about simple groups rather than indecomposable groups?

A first thing that we do to analyze something is "divide and conqur." So it is natural to consider simple groups. In the same way, it appears to be natural to consider indecomposable groups. However, ...
2
votes
2answers
47 views

Subgroup Lattices and Dimension

I apologize in advance in the case that this question is nonsensical. If the idea isn't clear, I can perhaps explain more below. In the fall I am taking an undergraduate abstract algebra course, and ...
22
votes
6answers
438 views

Generic elementary group theory problems.

This question is about generic group theory problems. here are examples for what I’m referring to: Prove that any group of order $p^2$, where $p$ is a prime, is abelian. Let $G$ be a ...
9
votes
1answer
169 views

Could we reasonably classify finite groups?

So I have been reading some work on $p$-groups, and I noticed a particularly disturbing sentence: "There is no hope of finding a finite set of invariants that will define every p-group up to ...
18
votes
1answer
420 views

Is there a geometric idea behind Sylow's theorems?

I have a confession to make: none of the proofs of Sylow's theorems I saw clicked with me. My first abstract algebra courses were more on the algebraic side (without mention of group actions and ...
12
votes
3answers
901 views

Choosing an advanced group theory text: concerns

In this question, An Introduction to the Theory of Groups by Rotman is recommended twice as a good second-course group theory text. However, after reading the reviews here, and seeing this pdf of ...
3
votes
0answers
83 views

References about finite group theory

In your opinion which are the best books regarding the theory of finite groups? I think that a wonderful one is "Finite Group Theory - Michael Aschbacher". Many thanks.
3
votes
0answers
95 views

A few questions about nonabelian cohomology of finite groups.

I apologize in advance if these questions are broad or basic. I tried to read about them at the Wikipedia, but everything is written in the language of category theory, in which I have had no formal ...
3
votes
2answers
248 views

Why are only the first four alternating groups are non-simple?

I know asking for intuition in math is a generally flawed approach, but can anyone give any reason why only the first four alternating groups are non-simple?
5
votes
3answers
344 views

Looking for texts in representation theory

I recently finished a course in representation theory, and while I learned a lot from it, I know that there's a lot more in the subject that I missed. For the course we used Fulton and Harris as a ...
14
votes
2answers
442 views

Why isn't there interest in nontrivial, nondiscrete topologies on finite groups?

A topology on a group is required to be compatible with the group structure (multiplication must be a continuous map $G\times G\to G$ and inversion must be continuous). I've only ever seen the ...
13
votes
2answers
649 views

Is finite group theory still a fruitful area of research?

A while back I was learning about finite groups in an algebra class. I mentioned to a friend that finite group theory might be an interesting area of research to pursue. She asked something along the ...
4
votes
5answers
202 views

Are there methods to well order a finite group in a meaningful way?

Can some finite groups be well ordered in a "meaningful" way? I mean, it is clear that we can trivially find a bijection between $\{1,...,n\}$ and a finite group $G$ with $n$ elements, but I am ...