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Questions about the branch of algebra that deals with groups.

4 votes
1 answer
380 views

Finite groups with bounded centralizers

Let $G$ be a finite group. For each $x\in G$, the centralizer $\mathbf{C}_G(x)$ must contain $\langle x\rangle$. QUESTION: What are some interesting results of the following form: Given some bound on …
2 votes
0 answers
140 views

A possible generalization of "homotopy" to study group actions of various kinds

This is a naive question about abstract homotopy theory by someone who knows nothing about it, except that it involves some generalization of the notion of "homotopy". If we think of $O(n)$ as acting …
18 votes
2 answers
1k views

The mysterious significance of local subgroups in finite group theory

EDIT 21/12: Even if there are no conclusive answers to these questions, I would very much like to know if anyone has noted and attempted to explain the mysterious significance of local subgroups: are …
2 votes
1 answer
291 views

Groups (not necessarily finite) with a given number of maximal subgroups

It is somewhat easy to see that a group $G$ with exactly one maximal subgroup $M$ must be cyclic: any element in $G\setminus M$ generates $G$. EDIT: @YCor pointed out in the comments that this argumen …
1 vote
0 answers
104 views

Closed collections of finite groups

Let $\mathcal{C}$ be a collection of (isomorphism classes of) finite groups with the following properties: If $G\in\mathcal{C}$ and $H$ is a homomorphic image of $G$, then $H\in\mathcal{C}$ If $G\in\ …
4 votes
0 answers
204 views

A different approach to proving a property of finite solvable groups

Edit: I'd be happy to hear any vague thoughts you might have, however far they may be from a complete solution! I asked this on math.stackexchange a couple of days ago, but it didn't attract any atten …
8 votes
2 answers
908 views

Nonisomorphic finite groups with isomorphic Sylow subgroups

The broad theme that underlies this question is: to what extent can the study of finite groups be reduced to the study of $p$-groups? I imagine that it is possible for a pair of nonisomorphic finite …
4 votes
0 answers
105 views

"Interpretation" of families of conjugate subgroups in a finite group

For a fixed prime $p$, the Sylow $p$-subgroups of a given finite group are all conjugate. Here are some more examples of situations in which we find that subgroups of a finite group defined by a certa …