Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

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$K$-theory spectrum of the category of finite groups

(I asked some people this question in person and got the answer "no", but wanted to see if the Internet had more to say)$\newcommand{\FinGrp}{\mathbf{FinGrp}}$ Way back in my first group theory ...
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Which finite solvable groups have solvable automorphism groups?

Is it possible to give a reasonable description of those finite solvable groups $G$ such that $A = {\rm Aut}(G)$ is also solvable? The central case to deal with is that in which $G$ is a $p$-group of ...
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Sufficient condition for complementation of abelian normal subgroup

Suppose that we are given a finite $p$-group $G$ and an abelian normal subgroup $A$ of $G$. The question I have is whether any sufficient conditions are known for $A$ to have a complement in $G$. From ...
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Smith normal form of conjugacy class actions

This question was inspired by Smith Normal Form of a Cayley Graph of the Symmetric Group. Let $\mathbb{Q}S_n$ denote the group algebra over $\mathbb{Q}$ of the symmetric group $S_n$. Identify a ...
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Permutation groups with diameter $O(n \log n)$

I suspect that many permutation puzzles can be solved in $O(n \log n)$ moves, which has led me to the following question/conjecture: Suppose that 1. $P_i$ for $i<k=O(1)$ are permutations on an $n$ ...
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Let $M_n$ be the rank 2 free metabelian group of exponent $n$. For which $n$ is $M_n$ center-free? The abelianization $M_n^{ab}\cong C_n\times C_n$, so the commutator subgroup $M_n'$ is a cyclic $(\... 0answers 400 views "A remarkable Moufang loop" The 1985 paper A simple construction of the Fischer-Griess monster group by Conway refers to an "in press" article, A remarkable Moufang loop, with an application to the Fischer group$Fi_{24}$, by ... 0answers 277 views An abstract zero-sum problem I would like to know whether the problem described below has appeared in the literature and/or whether similar questions have been studied. I would be very happy to find some references or, if none ... 0answers 99 views Tensor products of irreducible representations of$GL_{2}(\mathbb{F}_{q})$Throughout the post$G = GL_{2}(\mathbb{F}_{q})$where$q$is a prime power with the prime not being 2. Let$V_{1}$and$V_{2}$be cuspidal representations of$G$over$\mathbb{C}$. I can understand ... 0answers 243 views Does there exist a nontrivial perfect group with a "locally commuting" presentation? EDIT: I originally insisted that the perfect group in question be finite, however I now realize that I do not need this condition, only that the generators used in the presentation have finite order. ... 0answers 198 views Computing the number of elementary abelian p-subgroups of rank 2 in$GL_{n}(\mathbb{F}_{p})$Let$p$be a prime number, and let$\mathbb{F}_{p}$be a finite field of order$p$. Let$GL_{n}(\mathbb{F}_{p})$denote the general linear group and$U_{n}$denote the unitriangular group of$n\times ...
The commutator width of a group is the smallest $n$ such that every product of commutators is a product of $n$ commutators. My initial question was: Do there exist finite perfect groups with ...
A group $G$ is called $\frac{3}{2}$-generated if every non-trivial element is contained in a generating pair, i.e. \forall g \in G \setminus \{e \}, \ \exists g' \in G \text{ such that } \langle g,g'...