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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 …

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\ …

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 …

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 …

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 …

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 …

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 …

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 …