Questions on group theory which concern finite groups.

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2
votes
1answer
79 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 = \...
7
votes
1answer
114 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 ...
2
votes
0answers
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 $a_{ij}>b$, ...
11
votes
1answer
299 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 ...
5
votes
1answer
216 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 ...
4
votes
0answers
295 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 $Rep(G)$...
4
votes
0answers
289 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 ...
15
votes
4answers
863 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 ...
9
votes
0answers
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 \ ,...
3
votes
1answer
322 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 $, ...
7
votes
3answers
383 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 ...
9
votes
2answers
242 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 ...
3
votes
1answer
127 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 almost-...
5
votes
1answer
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 $x\in G$. ...
7
votes
0answers
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 ...
0
votes
0answers
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 ...
8
votes
1answer
342 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. ...
1
vote
0answers
122 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 ...
3
votes
1answer
161 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 $\...
16
votes
4answers
759 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 ...
35
votes
4answers
2k 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 ...
2
votes
1answer
191 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, $d$ ...
-2
votes
1answer
85 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 P|=|M|...
3
votes
1answer
139 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 ...
1
vote
1answer
303 views

Classification of finite group schemes over a field

What is known about the classification of finite group schemes over a field? By a finite group scheme I mean $Spec A$ where $A$ is a finite-dimensional algebra over a field. Is there a full ...
6
votes
2answers
357 views

How simple does a $\mathbb{Q}$-simple group remain after base change to $\mathbb{Q}_{\ell}$?

Of course the general answer to the question in the title is: not very simple. I could not think of a better title, so let me explain my question in more detail. I have a number field $E/\mathbb{Q}$, ...
0
votes
0answers
92 views

When is a group generated by three involutions two of which commute a Coxeter group?

Let's say the group is generated by three involutions $a$, $b$, and $c$ such that $ord(ab)= 2$, $ord(bc)=3$, and $ord(ac)=m$. Under which conditions is it isomorphic to the rank 3 Coxeter group $(2, 3,...
2
votes
4answers
192 views

Are the orders of the generators of a group and the product of pairs of thereof enough for this group to be isomorphic to a Coxeter group?

Let's say we have $n$ generators $x_1, x_2, \cdots, x_n$ along with the following facts concerning their orders: \begin{eqnarray*} ord(x_i) &=& 2 \text{ for } i = 1, 2, \cdots, n \\ ord(x_i ...
9
votes
0answers
197 views

On shifted symmetric power sums

The functions $p^*_k(x)=\sum_{i=1}^N ((x_i-i)^k-(-i)^k)$ are analogues of power sum symmetric functions, called shifted symmetric by Okounkov and Olshanski. Define $p^*_{(k_1,k_2,...)}=p^*_{k_1}p^*_{...
1
vote
1answer
96 views

embedding of finite groups into product

Our situation is following. Assume that we have free product $\star_{i<n} G_i$ each $G_i$ finite group and assume that we have normal subgroup $K$ such that composition of canonical embedding and ...
2
votes
1answer
115 views

What is a good program for matrix groups computations?

I need a computer program, to help me with some very basic group computations. Specifically, I want to know if some group generated by a few small matrices over a finite field is solvable. Is there a ...
4
votes
0answers
206 views

Non averaging sequences in finite groups

Let us say that a non averaging sequence in a group $G$ is a sequence $x_1, \dots, x_n$ such that $$x_i^2 \neq x_j x_k$$ for any indices $i, j, k$ such that two at least are distinct. My question is: ...
0
votes
0answers
64 views

Commutator subgroup of rotational symmetries of the hypercube

I would like to know which is the commutator subgroup of the group of rotational isometries of the $n$-dimensional cube. The group i am talking about is the subgroup of $\{ \pm 1 \} \wr S_n$ ...
2
votes
0answers
121 views

Classification of Automorphism set of a Regular graph

Let $A$ be the adjacency matrix of an $r$-regular graph $G$ with $n$ vertices (Not complete or cycle graph) . Also, let $Aut(G)$ be the set of all its automorphisms (i.e. set of permutation matrices)....
5
votes
2answers
440 views

What is the permutation group generated by those three given morphisms of the affine space $\mathbb{F}_q^3$?

Let $\mathbb{A}^3 = \mathbb{F}_q^3$. Consider the following three functions $\mathbb{A}^3\to\mathbb{A}^3$: \begin{eqnarray*} h: (x, y, z) &\mapsto& (x, y, xy - z) \\ u: (x, y, z) &\mapsto&...
1
vote
1answer
250 views

Generalization of a theorem of Steinberg

Steinberg has a beautiful theorem counting $F$-stable maximal tori in a reductive group. Here's the version of the result that you can find as Theorem 3.4.1 of Carter's Finite groups of Lie type: ...
8
votes
1answer
803 views

Which finite groups have no irreducible representations other than characters?

A classical result states that all the irreducible representations of a finite group over $\mathbb{C}$ are characters if and only if $G$ is abelian. I would like to know what happens if we consider a ...
1
vote
1answer
142 views

Involutive automorphisms of a finite abelian p-group

First, let $A$ be a finitely generated free abelian group, and $s$ an automorphism of order $2$ of $A$. Set $G=\{1,s\}$. Then we know that $A$ is a sum of indecomposable $G$-lattices $A_i$, where ...
6
votes
4answers
446 views

SO$(4)$ (& SO$(n)$) characterization?

I believe it is the case that any finite subgroup of SO$(3)$ (the $3 \times 3$ orthogonal matrices of determinant $1$) is either a cyclic group $C_n$, or a dihedral group $D_n$, or one of the groups ...
2
votes
1answer
131 views

Counting elements with certain word length in abelian groups

Given a (finite) abelian group $G = \langle S \mid R \rangle$, has the problem of counting the number of elements which can be expressed as a word (in $S$) of length $\leq k$ been studied? If so, ...
3
votes
1answer
185 views

Is there a way to find an efficient set of relations for presenting the subgroup generated by two matrices in $SL(2, q)$?

Given two elements $a, b \in SL(2, \mathbb{F}_q)$, is there a way to find an efficient presentation $$\langle x, y \mid \text{relations}\rangle$$ of the subgroup $\langle a, b \rangle$? My intention ...
2
votes
2answers
134 views

What are the 2-generated subgroups of the special linear group $SL(2, q)$ over a finite field?

What is the subgroup structure of the subgroups $\langle a, b\rangle$ where $a, b \in SL(2, q)$?
6
votes
1answer
238 views

Is every nonabelian finite simple group a quotient of a triangle group $(a,b,c)$ with $a,b,c$ coprime?

Here by triangle group $(a,b,c)$ I mean the group with presentation $$\langle x,y \;|\; x^a = y^b = (xy)^c = 1\rangle$$ In other words, for every finite simple nonabelian group $G$, do there exist ...
4
votes
1answer
544 views

Is there a structure theorem or group law for finite groups generated by two elements?

Say that $a, b \in G$ are two elements of a finite group $G$. Is there a structure theorem for the structure of $\langle a,b\rangle$? Is there a way to derive group laws for the group operation in the ...
3
votes
1answer
174 views

A representation of a finite group where every nonzero vector has a trivial stabilizer [duplicate]

What are the finite groups which admit a non-zero representation in char 0 where every non-zero vector has stabilizer equal to $\left<1\right>$? Cyclic groups of prime order is one obvious class....
5
votes
0answers
218 views

A class 3 group of order 243

Let G be a group of order $243=3^5$. We denote by $(G_i)$ its lower central series and assume that $G$ has class $3$ and that $|G:G_2|=|G_3|=9$. We assume moreover that the cubing map factors as a (...
51
votes
1answer
4k views

Why can't a nonabelian group be 75% abelian?

This question asks for intuition, not a proof. An earlier question, Measures of non-abelian-ness was thoroughly answered by Arturo Magidin. A paper by Gustafson1 proves that, for a nonabelian group, ...
0
votes
0answers
37 views

Is there a non-solvable integral fusion category of square-free dimension?

A finite group of square-free order is solvable (see here). You can find the definition for a solvable fusion category in this paper. Question: Is there a non-solvable integral fusion category of ...
2
votes
0answers
207 views

Does knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ sometimes offer a significant advantage in finding $r$?

Is there a cyclic group $\mathcal G$ with generator $g$ for which the discrete log problem is assumed to be hard, but knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ for random $r$ makes finding $r$ easy? ...
2
votes
2answers
197 views

Isotropic subspaces in a symplectic vectorspace over $GF(q)$

Let $V$ be a symplectic vectorspace of dimension $2n$, and $r\mid n$. Is this statement true?"There is an isotropic spread of $r$ dimensional subspaces in $V$". By an isotropis subspace I mean a ...