Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

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2 votes
0 answers
61 views

What Cayley graphs arise as nodes+edges from "nice" polytopes and when are these polytopes convex?

The Permutohedron is a remarkable convex polytope in $R^n$, such that its nodes are indexed by permutations and edges correspond to the Cayley graph of $S_n$ with respect to the standard generators, i....
0 votes
0 answers
66 views

Groups $P$ of order $p^5$ with $\Omega_1(P)=P$

I have been working with (particular) groups $P$ of order $p^5$. In fact, the ones that interest me the most are those that satisfy $$\langle x\in P\mid x^p=1\rangle=:\Omega_1(P)=P.$$ After a search ...
4 votes
1 answer
251 views

A pair of non-conjugate subgroups: a simple proof

$\DeclareMathOperator\SO{SO}$Set \begin{equation} \begin{aligned} \Gamma_1 &= \left\{ I_{6}, \; \gamma_1:= \left( \begin{smallmatrix} 0&1\\ 1&0 \\ &&0&1\\ &&1&0\\ &...
0 votes
0 answers
55 views

Finite $p$-groups of maximal class whose generators have order $p$

Let $G$ be a finite $p$-group of maximal nilpotency class that is not cyclic of order $p^2$. Then $G$ is $2$-generated, say $G=\langle a,b\rangle$. Is there a classification in the case when $a^p=b^p=...
4 votes
1 answer
203 views

Diameter of the "Masterball-puzzle" permutation groups by a kind of Cartier-Foata enumeration?

There is an wonderful blog post by Jordan S. Ellenberg SHOULD YOU BE SURPRISED BY THE DIAMETER OF THE NXNXN RUBIK’S GROUP?. Which explains how one can come to $N^2log(N)$ estimate of the diameter of ...
3 votes
0 answers
102 views

In how many ways does a Lie algebra decompose as an orthogonal direct sum of Cartans?

For a prime $p$, the Lie algebra $\mathfrak{su}(p)$ can be decomposed into an orthogonal direct sum of $p+1$ Cartan subalgebras as follows. Consider the clock and shift matrices — these are a pair of ...
3 votes
1 answer
225 views

Finite-maximal subgroups of orthogonal groups

I define a finite subgroup $H$ of a group $G$, finite-maximal if for any $g\in G\setminus H$, $\langle H,g\rangle$ is infinite. My question is now to find the finite-maximal subgroups of $\mathrm{SO}...
1 vote
0 answers
107 views

Classification of complex irreducible representations of $\mathrm{GL}_n(\mathbb{F}_q)$ [duplicate]

Is there a classification of complex irreducible representations of the group $\operatorname{GL}_n(\mathbb{F}_q)$, where $\mathbb{F}_q$ is a finite field with $q$ elements?
1 vote
1 answer
93 views

Irreducible subspaces in the space of functions on Grassmannian acted by $\mathrm{GL}_n(\mathbb{F}_q)$

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Gr{Gr}$Let $\mathbb{F}_q$ be a finite field with $q$ elements. Let $\Gr_{i,n}(\mathbb{F}_q)$ denote the Grassmannian of linear $i$-dimensional ...
2 votes
0 answers
83 views

Estimating the cardinality of the set of conjugacy classes of subgroups in a finite group of given order

1. Let $G$ be a finite group of order $n$. I need an estimate for the number $c$ of conjugacy classes of subgroups $D\subseteq G$. Note that any subgroup of $G$ contains $1_G$, and so the set of all ...
1 vote
0 answers
146 views

Which groups can be generated by a single conjugacy class?

How can we characterize the finite groups generated by a subset of a single conjugacy class? This post asks for well-known families of finitely generated groups generated by a single conjugacy class. ...
1 vote
0 answers
133 views

Bounds for the orders of second largest subgroups of $\mathrm{SL}_n(\mathbb F_q)$

$\DeclareMathOperator\SL{SL}$By Patton's thesis, except for a finite number of possibilities, the $(n-1, 1)$ parabolic subgroup, $P$ say, has the largest number of elements among all non-trivial ...
4 votes
2 answers
214 views

Order of abelian subgroup of the automorphism group of an abelian group

Suppose $A$ is a finite abelian group and $B\leq\operatorname{Aut}(A)$ is abelian. Is it possible for the order of $B$ to be strictly greater than the order of $A$? What if I additionally impose that ...
3 votes
1 answer
289 views

Number of conjugacy classes of pairs of commuting elements

Let $G$ be a finite group and denote by $r_G$ the number of conjugacy classes of pairs of commuting elements, i.e. the cardinality of the following set $$ A_G = \{ c(a_1,a_2) \ | \ a_1,a_2 \in G \text{...
11 votes
2 answers
697 views

How small can maximal subgroups be?

Given a finite group $G$, let $p(G)$ denote the number of prime factors of the order of $G$ (counting multiplicities). Does there exist a function $f: \mathbb{N} \rightarrow \mathbb{N}$ which grows ...
97 votes
19 answers
36k views

Collecting proofs that finite multiplicative subgroups of fields are cyclic

I teach elementary number theory and discrete mathematics to students who come with no abstract algebra. I have found proving the key theorem that finite multiplicative subgroups of fields are cyclic ...
-2 votes
0 answers
54 views

The direct product of two proper commuting subgroups of a non-abelian indecomposable finite group G can be equal to G?

Let G be a finite non-abelian indecomposable group. Let $G_1$ and $G_2$ be two proper subgroups of $G$. Assume that $G_1$ and $G_2$ commute, i.e., for every $g_1$ in $G_1$ and $g_2$ in $G_2$ we have $...
6 votes
1 answer
389 views

Finite groups and noncommutative algebraic geometry

DISCLAIMER: My relationship with noncommutative algebraic geometry is that of a curious, ignorant bystander. I confess that I know very little about noncommutative algebraic geometry, but I am ...
17 votes
2 answers
825 views

The sum (with multiplicity) of the cubes of irreducible character degrees of a finite group

Throughout $G$ is a finite, non-abelian group. $\DeclareMathOperator\Irr{Irr}\DeclareMathOperator\AD{AD}\DeclareMathOperator\cp{cp}\newcommand\card[1]{\lvert#1\rvert}$ Let $\Irr(G)$ be the set of ...
7 votes
0 answers
181 views

Growth of spheres in FINITE nilpotent groups - Gaussian approximation (central limit theorem)?

Standard setup. Consider a group and choose generators. Word-metric (or in the other words - distance on the Cayley graph of the group+generators) - converts a group into a metric space, which is ...
2 votes
0 answers
84 views

Are the alternating pentultimate and the $2$-alternating pentultimate isomorphic?

Consider a physical puzzle which is in the shape of a dodecahedron where the pieces that move are the face centers and you can slice it through a plane parallel to two opposite faces going through the ...
4 votes
0 answers
186 views

Polynomials of growth for finite Heisenberg groups

Take a standard finite Heisenberg group with two standard generators and let's consider its growth polynomial - the polynomial which coefficients are equal to the sphere sizes. For example for $H_3(Z/...
6 votes
2 answers
398 views

Embedding $\mathrm{SL}_n(3)$ into $\mathrm{SL}_n(p)$

$\DeclareMathOperator\SL{SL}$Let $p$ be an odd prime. It is easy to show that $\SL_2(3)$ can be embedded into $\SL_2(p)$. Now, let $n$ be an integer larger than $2$. Question: In which circumstances, $...
3 votes
1 answer
262 views

Can we relate the character of the permutation representation of $G$ on the cosets $G/\langle g_i\rangle$ to the number of cycles of $g_i$?

Let $G$ be a finite group generated by permutations $g_1,\dots,g_s$ such that $g_1g_2\cdots g_s$ is the identity permutation. The corresponding Hurwitz representation $V_{\text{Hur}}$ has character $$\...
64 votes
2 answers
8k views

Where are the second- (and third-)generation proofs of the classification of finite simple groups up to?

According the the Wikipedia page, the second generation proof is up to at least nine volumes: six by Gorenstein, Lyons and Solomon dated 1994-2005, two covering the quasithin business by Aschbacher ...
0 votes
0 answers
56 views

Involutions in $\operatorname {PSO}(4,K)$

In the algebraic group $G=\operatorname {PSO}(4,K)$ where $K$ is an algebraically closed field of an odd characteristic, how many different classes of involutions are there and what are the ...
1 vote
0 answers
156 views

Applications of Artin's theorem on induced representations

Let $G$ be a finite group and let $R(G)$ be the (complex) representation ring of $G$. As stated in Serre's book on representation theory, Artin's theorem says the following: Theorem: Let $X$ be a ...
2 votes
3 answers
858 views

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 ...
4 votes
3 answers
551 views

Regular orbits for automorphisms of finite simple groups

Let $G$ be a finite group and $f$ be an automorphism of $G$. We say that $f$ has a regular orbit if there exists $x\in G$ such that $|x^f|=|f|$. If $G$ is abelian it is known that every automorphism ...
14 votes
0 answers
472 views

Is the monster group maximal in SO(196883)?

$\DeclareMathOperator\SO{SO}$The smallest degree of a nontrivial complex representation of the monster group $ M $ is $ 196883 $. This irrep has Schur indicator $ 1 $, so the image must lie in the ...
7 votes
1 answer
620 views

What are double groups mathematically?

In physics and chemistry, there is the concept of double groups. These are double covers of the usual point groups, obtained by "adding an element $R$, which represents rotation by $2\pi$" ...
3 votes
0 answers
98 views

Exact structures on representations of a finite group

For simplicity assume $G$ is a (finite) $p$-group, and $k$ is field of characteristic $p$, so that there exists a unique simple $kG$-module the trivial module $k$. I am looking for a class of short ...
3 votes
1 answer
223 views

Nonisomorphic central products on the same pair of groups?

A central product of two groups $G$ and $H$ is determined as follows. The groups $G$ and $H$ have respective central subgroups $A$ and $B$ which are isomorphic, let $\delta:A\rightarrow B$ be such ...
14 votes
1 answer
746 views

On the iterated automorphism groups of the cyclic groups

Let $C_n$ be the cyclic group of order $n$. Its automorphism group $Aut(C_n)$ is a group of order $\varphi(n)$ isomorphic to $(\mathbb{Z}/n\mathbb{Z})^{\times}$ the multiplicative group of integer ...
3 votes
1 answer
261 views

Distinct characters with the same character values, outer automorphisms and Galois conjugation

Given an (irreducible complex) character of a finite group the following three construction all yield another irreducible character of the same degree: multiplying by a degree 1 character applying an ...
5 votes
2 answers
673 views

Factor subsets of a finite group

Let $G$ be a group of order $n$ and $d$ a positive divisor of $n$. Is it true that there exists a subset $A$ of $G$ with $d$ elements and a subset $B$ such that $G=AB$ and $|AB|=|A||B|$ (equivalently,...
0 votes
1 answer
94 views

An explicit matrix form in the symplectic group

In the algebraic group $G=\operatorname {PCSp}(2^{r},K)$ where $K$ is an algebraically closed field of an odd characteristic, there is a conjugacy class of involutions with representative: $$ e=\left[...
0 votes
1 answer
83 views

An explicit matrix form

In the algebraic group $G=\operatorname {PCGO}(2m,K)$ where $K$ is an algebraically closed field of an odd characteristic, there is a conjugacy class of involutions with representative: $$ e=\left[ \...
1 vote
1 answer
332 views

Distribution of 2-groups

In the family of finite groups of order less than $2000$, there are about 99% of order $1024$, so I have a question about $2$-groups: Let $f(n)$ be the number of non-isomorphic finite groups of order $...
6 votes
0 answers
114 views

Sylow subgroups of the restricted Burnside group $\mathrm{RB}(d,n)$?

$\DeclareMathOperator\RB{RB}$What is known about the Sylow subgroups of the restricted Burnside groups $\RB(d,n)$ ? I am looking for a reference. In fact my question is slightly more general. Recall ...
6 votes
1 answer
210 views

Classification of non-abelian simple groups with cyclic T.I. Sylow p -subgroup

Let $G$ be a finite group and $H$ a subgroup of $G$. We say that $H$ is a trivial intersection (for short T.I.) subgroup of $G$ if $H\cap H^x=1$ for each $x\in G-N_G(H)$. I read the next result in the ...
1 vote
0 answers
114 views

Irreducible projective representations of finite abelian groups

I want to know if there is a description of all irreducible complex projective representations of an arbitrary finite abelian group. I have seen this for particular cases such as those given here and ...
4 votes
0 answers
292 views

Are there infinitely many simple integral fusion rings of rank $4$?

$\DeclareMathOperator\ch{ch}$$\DeclareMathOperator\FPdim{FPdim}$We refer to [EGNO15, Chapter 3] for the notion of fusion ring and basic results. The type of a fusion ring $R$ is the list $(\FPdim(b_i)...
2 votes
0 answers
124 views

Automorphisms of (nilpotent) groups : torsion cokernel on the abelianisation implies torsion cokernel on the center?

$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\End{End}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Z{Z}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\Span{Span}\DeclareMathOperator\ker{ker}\...
4 votes
1 answer
353 views

Finite groups with bounded centralizers

Let $G$ be a finite group. For each $x\in G$, the centralizer $\mathbf{C}_G(x)$ must contain $\langle x\rangle$. QUESTION: What are some interesting results of the following form: Given some bound on $...
2 votes
0 answers
79 views

Splitting of $\mathbb{Z}/p\to E\to (\mathbb{Z}/p)^n$ in cohomological terms

Let $d>1$ be an odd integer. Given a simplicial set $X$ and $[\gamma]\in H^2(X,\mathbb{Z}/d)$, there exists a fibration $N\mathbb{Z}/d\to E\to X$, with $$E= X_\gamma:=N\mathbb{Z}/d\times_{\gamma} X....
5 votes
1 answer
334 views

Number of $k$-tuples of elements generating a cyclic group

Let $k$, $m$ be natural numbers, and $C_m:=\mathbb{Z}/ m \mathbb{Z}$ be the cyclic group of order $m$. Let $N_{k, \, m}$ be the cardinality of the following set: $$\{(a_1, \ldots, a_k) \in (C_m)^k \; ...
1 vote
0 answers
45 views

Sylow subgroups of the free product of profinite groups

I am interested in the Sylow subgroups of the profinite completion of a free product of finite groups. Is the following naive expectation true ? I assume things like this should be well-known, and am ...
8 votes
1 answer
353 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. ...
20 votes
1 answer
942 views

Proof of CFSG assuming every simple group is two-generated

It is well-known that one of the corollaries of the classification of finite simple groups (CFSG) is that every finite simple group can be generated by two elements. In a comment on an answer to an ...

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