# Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

571 questions with no upvoted or accepted answers
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### Is there a "direct" proof of the Galois symmetry on centre of group algebra?

Let $G$ be a finite group, and $n$ an integer coprime to $|G|$. Then we have the following map, which is clearly not a morphism of groups in general: $$g\mapsto g^n.$$ This induces a linear ...
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• 1,294
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### What properties characterize the function $L(x) = x+\exp(x) \log(x)$?

As might be known, the function $L(x) = x+\exp(x)\log(x)$ plays a prominent role in the Lagarias formulation of the Riemann hypothesis: $$\sigma(n) \le H_n + \exp(H_n) \log(H_n)$$ My question is, ...
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### Cyclic and prime factorizations of finite groups

A tuple $(A_1,\dots,A_n)$ of subsets of a finite group $G$ is called a factorization of $G$ if $G=A_1\cdots A_n$ and $|A_1|\cdots|A_n|=G$. In Cryptology factorizations of groups are known as ...
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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 ...
• 1,483
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### Detecting a module for the free group algebra on a finite quotient

Let $F_2$ be the free group on two generators $x,y$ and let $R$ be the group algebra $\mathbf{Q}[F_2]$. Let $a,b,c$ be integers. Then define a right $R$-module $M = R / (ax + by + c) R$. I am ...
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### Is every finite group the automorphism group of a smooth projective curve?

$\DeclareMathOperator\Aut{Aut}$Let $G$ be a finite group and let $k$ be a field with algebraic closure $K$. Is there a smooth projective curve $C$ defined over $k$ such that $\Aut_k(C)=\Aut_K(C)$ is ...
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### Is the norm element characteristic in modular group rings?

Let $G$ be a finite $p$- group and let $\varphi$ be an automorphism of $\mathbb{F}_pG$ as $\mathbb{F}_p$-algebras and let $n = \sum_{g\in G} g$ be the norm element. Does it follow that $\varphi(n)=n$? ...
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### 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 ...
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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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### Breuer-Guralnick-Kantor conjecture and infinite 3/2-generated groups

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'...
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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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