Questions on group theory which concern finite groups.

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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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**2**answers

352 views

### 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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**2**answers

185 views

### 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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90 views

### 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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**1**answer

104 views

### Finding a semigroup that maximizes the trace of a sum of matrices

Let $H$ be a finite semigroup containing $n$ elements from a compact group $G$. I am trying to solve
$$\max_{h_i,\ h_j\ \in\ H} \operatorname{tr} \sum_{i,\ j\ \leq\ n} \rho(h_i^{-1} h_j)(A_j ...

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66 views

### Salvaging Howson's theorem for free profinite groups

This is an attempt to find a correct version of Do free profinite groups satisfy Howson's theorem?
Let $F$ be a free profinite group, and let $A,B \leq F$ be closed finitely generated subgroups ...

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316 views

### Why does iterated indexing avoid cycles of length 5?

Start with a permutation $s_0$ of the numbers
$(1,\ldots,n)$, e.g., for $n=10$,
$s_0=(8,2,1,6,9,7,10,5,4,3)$.
Form $s_1$ by using the numbers in $s_0$ as indices into $s_0$.
So $s_1$ is composed of ...

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**1**answer

173 views

### Existence of some character degree in a solvable group

Let $p=2^n-1$ be a Mersenne prime. We know that $H=PSL(2,p^4)$ has an irreducible character of degree $ p^4$.
Is there any solvable group of order $H$ with an irreducible character of degree $ p^4$. ...

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**1**answer

77 views

### recovering information about a group from the spectrum of its Cayley graph

Suppose you have a finite group and you consider its Cayley graph with respect to some fixed generating set of nonidentity elements closed under inversion. Are there any results known to the effect ...

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289 views

### Faithful projective representations of symmetric groups

This is a reference request.
Do you know where I can find the dimensions of the faithful projective representations of $S_n$ and $A_n$ for $n\ge 5$?
Thank you in advance.

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**3**answers

537 views

### Beyond Brauer's theorem

Brauer's classic theorem states that any character of a finite group can be expressed as a linear combination with integer coefficients of characters induced by linear characters of p-elementary ...

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**1**answer

642 views

### “Squeezing” a finite group between symmetric groups

For a finite group $G$, take $m$ as the largest integer such that $G$ has a subgroup $H\cong S_m$ and $n$ as the smallest integer such that $G$ is itself isomorphic to a subgroup of $S_n$. We then ...

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390 views

### Counting cyclic subgroups of order $p^{2}$: $p$ an odd prime vs. $p=2$

While playing with Frobenius' problem (about finite groups $G$ in which, for some positive integer $n \mid |G|$, there are exactly $n$ elements of order dividing $n$), I came up with the following ...

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**1**answer

163 views

### Intersection of maximal subgroups of PSL(2,q)

Let $G := PSL(2,2^r)$, and let $M$ be a maximal subgroup of $G$ isomorphic to $PSL(2,2^s)$. I need to compute $H := M \cap M^g$ for $g \in G-M$. It seems to me that $|H|$ must be $2^r, 2^r\pm 1$ or ...

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932 views

### Number of squares in a finite group

This was asked at MSE but never answered.
Let $G$ be a finite group and denote by $sq(G)$ the number of squares in $G$ i.e. the number of elements in $G$ which possess a square root. For example, if
...

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240 views

### The number of subgroups of ${\frak S}_n$

Because of my interest in this question, I listed the subgroups of ${\frak S}_n$ for $1\le n\le4$. I found that the number of subgroups are, respectively, $1,2,6,24$. It might be a coincidence, or it ...

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votes

**1**answer

158 views

### Reference sought regarding the fact that all non-abelian finite simple groups are 2-generated

I've learned that all non-abelian finite simple groups are $2$-generated,
i.e. have a generating set of cardinality $2$.
Is there a reference to this statement which does not just point to the
...

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votes

**1**answer

108 views

### positions of regular cubes in Euclidean space with all its vertices without distinction

Let $P$ be the standard regular cube, centered at the origin of $\mathbb{R}^3$ with all its vertices on the unit sphere.
If the vertices are labelled by $1,2,3,4,5,6,7,8$, then the collection of ...

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146 views

### Which Dihedral Groups are $\text{CI}$-Groups?

Let $D_{n}$ denotes the dihedral group of order $2n$. Firstly, for self-referencing of the question, I give some definitions which are standard.
Let $G$ be a finite group. A subset $S$ of group $G$ ...

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**1**answer

177 views

### Structure of Deligne-Lusztig representations $R_{T,\theta}$ for ministropic $T$ and cuspidal representations

Let $G$ be a reductive group over a finite field $k$, let $F$ be a Frobenius morphism on $G$.
I'll start with a somewhat vague question and make my question more specific further down:
How do ...

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71 views

### maximal elementary abelian p-subgroups of finite groups of Lie type

Let $G$ be a simple algebraic group over an algebraically closed field k of characteristic $p$. Let $G(\mathbb{F}_{p^r})$ be the corresponding finite group of Lie type. Is it true that every maximal ...

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69 views

### associativity of the extension of finie groups [closed]

Following my previous question I have two questions:
1.the extensions of the group is associative or not, i.e. as we know by the notations of atlas if $S={\rm PSL}(2,q)$, where $q=p^f>9$, then ...

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**1**answer

148 views

### Extensions of $\Bbb Z_3$ by $PGL(2,q)$ where $q$ is odd

Let $q$ be odd. If $G$ is a finite group such that $G$ has a normal subgroup $H$ of order $3$ such that $G/H\cong {\rm PGL}(2,q)$, what can we say about $G$. Is it true in general that $G\cong {\Bbb ...

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279 views

### mod p cohomology ring of alternating groups

Let $A_n$ be the alternating group of $\{1,2,\cdots,n\}$.
(1). What is the cohomology ring
$$
H^*(A_4;\mathbb{Z}/3)
$$
and its Steenrod operation $P^i$'s?
(2). Are there general results about the ...

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109 views

### Exotic 2-adic lifts of mod $2$ Steinberg idempotent

Denote $B_n$ the Borel subgroup of $Gl_n(Z/2)$, i.e., the subgroup of
upper triangular matrices, $\Sigma _n$ the subgroup of permutation matrices.
The (conjugate) Steinberg idempotent is defined to be ...

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218 views

### To calculate $Tor_1^G(\mathbb{Z},N_{ab})$ and $Tor_1^Q(\mathbb{Z},N_{ab})$

Let $G$ be a finite $p$-group and $N$ be a normal subgroup of $G$. I wish to calculate $Tor_1^G(\mathbb{Z},N_{ab})$ and $Tor_1^Q(\mathbb{Z},N_{ab})$, where $Q=G/N$. I could not found any lecture notes ...

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**1**answer

77 views

### The maximal possible rank of a subgroup of a product of special linear groups

In this question I ask for a generalization of What is the maximal possible rank of a subgroup of a special linear group mod a prime?
Let $p_1, \dots, p_r$ be $r$ distinct odd primes.
Set $$G = ...

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votes

**1**answer

110 views

### 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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53 views

### 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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296 views

### 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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**1**answer

201 views

### 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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288 views

### 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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286 views

### 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 ...

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**4**answers

806 views

### What are the possible automorphism groups of Riemann surfaces of low genus?

Skipping the easy cases of genus 0 and 1, what groups can arise as the group of conformal transformations of a Riemann surfaces of genus, say, 2 or 3?
I'm frustrated because there are papers that ...

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182 views

### Is an inclusion of finite groups with boolean lattice, linearly primitive?

Let $(H \subset G)$ be an inclusion of finite groups.
Definition: Let $W$ be a representation of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{H}:=\{w \in W \ \vert \ kw=w \ ...

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**1**answer

302 views

### Maximal ideal of group ring

Let $R$ be a finite commutative ring with identity and $G$ an finite abelian group. Is there any more conditions (on $R$ or on $G $) under which we can characterize maximal ideals of group ring $RG $, ...

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365 views

### Computations in modular cohomology of finite groups

Let $k$ be an algebraically closed field of characteristic $p$, let $G$ be a finite group whose order is divisible by $p$, and let $H(G)$ be the commutative cohomology algebra of $G$ with coefficients ...

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240 views

### Linear occurrences of finite simple groups

Let $S$ be a finite simple group. All representations below are over the complex numbers.
Let
$d_0(S)$ be the smallest dimension of a faithful representation of $S$,
$d_1(S)$ be the smallest ...

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votes

**1**answer

125 views

### Character degrees of extensions of 2^B_2(q^2)

The outer automorphism group of the Suzuki simple group
${}^2B_2 (2^{2m+1})$, $m \geq 1$ is cyclic of order $2m+1$ and
is generated by a field automorphism $\varphi$ of order $2m+1$.
For any ...

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**1**answer

188 views

### Generalized identities of (soluble) groups

Let $G$ be a group. Let us say that $G$ satisfies a generalized identity of degree $n$ if there exist $a_1,a_2,\dots a_n \in G$ such that
$$x^{a_1}x^{a_2}\dots x^{a_n}=1,$$
for all $x\in G$.
...

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123 views

### Rings that are $K_0$ of finite groups

Is there a simple characterisation of all rings which appear as $K_0$ of finite groups? By $K_0$ of a finite group $G$ I mean $K_0(\mathbb C[G])$ which in the same as a ring of virtual characters of ...

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53 views

### Equivalence classes of pairs linear transformations

Consider the set of 4-tuples:
$$S_{(x, y), k} = \{ (a_i, b_i, a_j, b_j) : \|a_ixb_i - a_jyb_j\|_F^2 = k \}$$
for $a \in GL(m, \mathbb{R})$, $b \in GL(n, \mathbb{R})$, $x, y \in \mathbb{R}^{m \times ...

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**1**answer

339 views

### Three involutions on the set of 6-box Young diagrams

The set of $n$-box Young diagrams classifies both conjugacy classes in $S_n$ and equivalence classes of irreducible representations of $S_n$. There is an outer automorphism of $S_6$, of order 2. ...

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119 views

### Discrete group action on the sphere

Let $f$ be a continuous function on $S^3$ and let $\xi^{\perp}=\{x\in S^3:\,x\cdot\xi=0\}$
be a two-dimensional equator of $S^3$ orthogonal to the direction $\xi\in S^3$ (here $x\cdot\xi$ stands for a ...

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**1**answer

150 views

### Center of an irreducible representation over $\mathbb{Q}$

Let $G$ be a finite group, $\rho\colon G \rightarrow \mathrm{GL}_n(\mathbb{Q})$ its irreducible representation, and $D$ the division algebra of $G$-endomorphisms of $\mathbb{Q}^n$. The division ...

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740 views

### What algebraic structures are related to the McGee graph?

Recall that an $(n,g)$-graph is a simple graph where each node has $n$ neighbors and the shortest cycle has length $g$, while an $(n,g)$-cage is $(n,g)$-graph with the minimum number of nodes.
The ...

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**3**answers

1k views

### How much of the ATLAS of finite groups is independently checked and/or computer verified?

In a recent talk Serre made some comments about proofs that rely on the classification of finite simple groups (CFSG) and on the ATLAS of Finite Groups. Namely, he said that a proof that relied on the ...

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**1**answer

188 views

### Finite sequences which happen to be the sequence of orders of elements of a simple group

Let $n\in \Bbb N$. Any finite group $G$ with $|G|=n$ has a solution for the Diophantine equation
$$\sum_{d|n}x_d\phi(d)=n~~~~~~~~~~~~~~~~~~~~~~~(1)$$
where $\phi$ is the Euler's totient function, $d$ ...

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**1**answer

83 views

### On Sylow subgroup of a finite group [closed]

Let $p\mid n$, then by $n_p$ we mean the $p$-part of $n$, i.e. $n_p = p^k$ if $p^k\mid n$ but $p^{k+1}\nmid n$. Let $G$ be a finite group, $M\leq G$ and $P\in Syl_p(G)$. Is It true that $|M\cap ...

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133 views

### Classification of finite abelian hypergroups and table algebras

Update: Originally, I formulated this question for finite abelian hypergroups, but in a discussion with Geoff Robinson below I realized that the abelian hypergroups defined below are equivalent to ...