# Tagged Questions

Questions on group theory which concern finite groups.

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### The number of submodules of $\mathbb{Z}_q^n$

Observe $\mathbb{Z}_q^n = \mathbb{Z}_q \times \cdots \times\mathbb{Z}_q$ as a module over $\mathbb{Z}_q\equiv\mathbb{Z}/q\mathbb{Z}$, for general $q$. I am interested in the following questions: How ...
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### Determining conjugacy class of a subgroup from the sizes of its intersections with the conjugacy classes

Let $C_1,C_2,\ldots,C_n$ be the conjugacy classes of a finite group $G$. It is possible that there are two non-conjugate subgroups $K \leq G$ and $H \leq G$ such that $|H \cap C_i|=|K \cap C_i|$ for ...
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### Expository papers for Feit–Thompson Theorem [duplicate]

Feit–Thompson theorem states that every finite group of odd order is solvable. Its proof is near 300 pages so it is definitely not an easy paper to read without a prior knowledge about the general ...
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### 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 ...
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### Centerless finite groups with exactly three prime divisors

Let $G$ be a centerless finitee group and $\pi(G)=\{p,q,r\}$ be the set of prime divisors of $|G|$. I have some questions about these groups. Is there any classification of such groups? In what ...
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### Is the Frattini subgroup of a free profinite group trivial?

Let $\widehat{F_2}$ be the free profinite group of rank 2. Is its Frattini subgroup $\Phi(\widehat{F_2})$ trivial? I know that the Frattini subgroup of a pro-$p$ group is open. On the other hand, the ...
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### Representations of orthogonal groups over the field of two elements

I am looking for some references on modular representation theory of the orthogonal groups $O_{2n+1}(2)$, $O_{2n}^{+}(2)$, or $O_{2n}^{-}(2)$ over $\mathbb{F}_2$.
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### Minimal maximal subgroup of the symmetric group

The question is pretty much in the title: What is the maximal subgroup of $S_d$ of maximal index (so minimal size)? A slight variant (I am not sure if it leads to a different answer) is: what if we ...
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### Rank gradient in free products amalgamating a finite subgroup

Let $A,B$ be finitely generated groups with a common finite subgroup $C$. Suppose that $[A : C] > 2, [B : C] > 1$. Must $A *_C B$ have positive rank gradient? See Which 3-manifolds have ...
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### Characterization of finite groups using sum of the orders of their elements

Notation: If $G$ is a finite group, $o(g)$ denotes the order of the element $g\in G$. Motivation: Some finite groups could be uniquely determined by the size of the group. For example given a prime ...
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### a question about a group which has a factor isomorphic to a permutation group

Thanks for any help or comments. Suppose $G$ is a finite group and $t$ is an involution in $G$. Suppose also that $\frac{C_G(t)}{\langle t\rangle}\cong \mathbb{S}_{n-2}$ and $C_G(t)$ contains some ...
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### When can the restriction map be zero or onto?

Let $G$ be a finite $p$ group of order $p^n$ and $N$ be a subgroup of $G$ of index $p$. Let $res:H^2(G,A) \rightarrow H^2(N,A)$ be the restriction map on cohomology, where $A$ is a trivial $G$ module. ...
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### What is the maximal possible rank of a subgroup of a special linear group mod a prime?

Let $p$ be a prime number, and let $\mathbb{F}_p$ be the unique field of cardinality $p$. What is $\max \{d(H) : H \leq \mathrm{SL}_3(\mathbb{F}_p)\}$? Here we denote by $d(G)$ the smallest ...
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### Integral of a parametrized commutator

I am trying to solve the following integral $$\int_{-1}^{1}\;db\;||[t_{b}(A),J]||_{F}^{2}$$ where $t_{b}$ is the entrywise threshold of the matrix A ($0$ if $a_{ij}<b$, $a_{ij}$ if $a_{ij}>b$, ...
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### Elementary consequences of commuting limits and colimits over groups

This is a crosspost of this MSE question. In this n-cat cafe post, it is proven that for finite groups $G,H$ of coprime order, $G$-colimits and $H$-limits commute. Later on the following theorem is ...
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### Maximal subgroups of special linear groups over finite fields

Let $p$ be a prime number, and denote by $\mathbb{F}_p$ the field with $p$ elements. Is there a classification of the maximal subgroups of $G = \mathrm{SL}_3(\mathbb{F}_p)$ ? I am interested in ...
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### Euler characteristic, character of group representation and Riemann Roch theorem

I am considering the following set up:Let $G$ be a finite group,let $Rep(G)$ denote the category of finite dimensional representations over $\mathbb{C}$. Let $V,W$ be representations of $G$ in ...
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### Characterization for special linear group over finite fields

Thanks for any help or comments. In my research I need a characterization for special linear groups over finite fields by some information of its subgroups, especially centralizers. I saw the ...