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Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

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2 votes
0 answers
61 views

Classification of centralizers of elements of finite simple groups of Lie type

I am currently studying the twisted Ree finite simple groups given by $^2G_2(3^{2n+1})$ and I was wondering if there is a reference for the classification of centralizers of elements in this family of ...
5 votes
1 answer
368 views

Boolean ring of unitary divisors / Structure of unitary divisors?

I hope this question is appropriate for MO: Let $n$ be a natural number, $U_n := \{ d | d \text{ divides } n, \gcd(d,n/d)=1\}$ be the set of unitary divisors. We can make $U_n$ to a boolean ring: $$a \...
1 vote
1 answer
262 views

Group element of group algebra

For a prime $p$, let $G$ be a finite $p$-group and $F_{p}$ the field with $p$ elements. Let $A=\{a\in F_{p}G \mid a^{\sum_{x\in G}x}\neq 0\}$, where $F_pG$ is the group algebra of $G$ over $F_p$ and $...
12 votes
0 answers
340 views

Does every finite group have a small projective representation (over some ring)?

Question. Let $G$ be a finite group. Can we find some (commutative) ring $R$ and some positive integer $d=O(\log\lvert G\rvert)$ such that $G$ can be found as a subgroup of $\operatorname{PGL}_d(R)$? ...
5 votes
1 answer
653 views

What are the maximal closed subgroups of $ \operatorname{SU}_3 $?

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\SL{SL}$What are the maximal closed subgroups of $ \SU_3 $? This question is inspired by Lie subgroups of SU(3). Interesting partial answers to that ...
2 votes
2 answers
206 views

Software library for complex irreducible representations of $\mathrm{PSL}_n(q)$

I came across an extremely useful Python software library for the Monster group: https://github.com/Martin-Seysen/mmgroup which allows for all sorts of manipulations involving the sporadic finite ...
2 votes
0 answers
163 views

Nonabelian groups where every element has small order

Let $G$ be a finite nonabelian group with the property that if $g \in G$, then $$\DeclareMathOperator{\ord}{ord} \ord(g) \leqslant 10 \log_2 |G|, $$ where $\ord(g)$ is the order of the element $g$, ...
0 votes
1 answer
65 views

Groups with $2$-transitive permutation representations of different degrees

Suppose $G$ is a finite group, and suppose that it acts $2$-transitively in each of the permutation representations $(G,X_i)$ ($i$ ranges over some index set $I$), where the $X_i$s all have different ...
5 votes
1 answer
364 views

Groups with no proper non-trivial fully invariant subgroup

Let $G$ be a finite group. A subgroup $H$ of $G$ is said to be characteristic if $\phi(H)\subseteq H$, $\forall \phi \in \operatorname{Aut}(G)$ and fully invariant if $\phi(H)\subseteq H$, $\forall \...
5 votes
0 answers
85 views

Application of character sheaves to characters of $G(\mathbb{F}_q)$

I wish to ask about good examples of new applications of Lusztig's theory of character sheaves (and subsequent development, but excluding generalized Springer theory) back to the theory of characters ...
2 votes
0 answers
100 views

Finite groups of Lie type

Table $22.1$ Finite groups of Lie type from "Linear algebraic groups and finite groups of Lie type" by Malle and Testerman: For the type $A$: $G_{sc}^{F}=\operatorname{SL}_{n}(q)$ and $G_{ad}...
1 vote
1 answer
80 views

What are the efficient algorithms to compute Hamiltonian paths on Cayley graphs of finite groups ? Can GAP do it?

The famous Lovasz conjecture predicts existence of the Hamiltonian path on Cayley graphs. In general finding such a path is NP-complete problem, but there are many heuristic algorithms. Question 1: ...
9 votes
0 answers
292 views

Tilings in finite (not necessarily Abelian) groups

Let $G$ be a finite (not necessarily abelian) group. We call $A \subseteq G$ a right-tiling (for simplicity, a tiling) of $G$ if there exists a $B \subseteq G$ so that $$ G = \bigsqcup_{b\in B} bA.$$ ...
8 votes
1 answer
1k views

GAP cannot solve Rubik's cube 4x4x4 and higher ? (Practical limits of Schreier–Sims algorithm)

According to our practical experiments and literature search - computer algebra system GAP cannot "solve" Rubik's cube 4x4x4 and higher. That means cannot decompose given random element of ...
4 votes
0 answers
180 views

Subgroups that conjugate-cover the ambient group

Let $G$ be a finite group, and suppose that a set of proper subgroups $H_1,\dotsc,H_n$ satisfy $G=\bigcup_{g\in G}\bigcup_{i=1}^nH_i^g$, where $H_i^g$ is the conjugate of $H_i$ by $g$. In this case, ...
4 votes
1 answer
161 views

Finite group is Dedekind iff for every irrep, every element acts as identity or has all eigenvalues $\ne 1$

Consider the following claim: a finite group $G$ is Dedekind $\iff$ for every irrep $\rho$, and every $g \in G$, $\rho(g)$ either is identity matrix or has all eigenvalues $\ne 1$. Is this claim true? ...
2 votes
1 answer
161 views

Smallest dimensional faithful complex representation of $\mathrm{PSL}(k,q)$

For given $k>1$ and $q$ a prime power, what is the minimal dimension, as a function of $(k,q)$, for which a faithful complex representation of the projective special linear group over $\mathbb{F}_q$...
4 votes
1 answer
441 views

Large(ish) finite non-abelian subgroups of $\operatorname{GL}_n \mathbb C$ for $n>70$

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\SU{SU}\newcommand{\C}{\mathbb{C}}$My question is about large order finite non-abelian subgroups of $\GL_n\C$ without an ...
4 votes
0 answers
115 views

Complexity to find "short" (e.g. polynomial in diameter) decomposition of the permutation into the product of generators?

Question 1: Consider the symmetric group $S_n$ and some set of permutations $p_i$. Given permutation $g$ - what is known about the algorithmic complexity to decompose $g$ into product of $p_i$ ...
3 votes
0 answers
89 views

Which elements in $\mathrm{Aut}(\widehat{F_2})$ preserve the procyclic subgroup generated by the commutator $c=[a,b]$?

Let $F_2$ denote the free group over two generators $a,b$, and we denote $c=[a,b]$ as the commutator. It is well-known that any automorphism $\psi$ of $F_2$ preserves the conjugacy class of the ...
2 votes
0 answers
167 views

Centralizer of PSL in PGL and of SL in GL: reference request

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\PSL{PSL}$Consider the general linear group $\GL(n,q)$ over a finite field with $q$ elements and ...
10 votes
0 answers
424 views

Function related to length of group presentations: is it computable?

(This question comes from a friend who works in sofic group theory.) Consider the function $f: \mathbb{N} \to \mathbb{N}$, defined, for any $n \in \mathbb{N}$, by putting $f(n)$ to be the largest ...
10 votes
2 answers
337 views

Finitely dominated universal spaces for the family of solvable subgroups

$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Sz{Sz}$In short, I am interested in the question which finite groups $G$ admit a finitely dominated universal space with respect to the family of ...
3 votes
0 answers
51 views

Asymptotic dimension of graph families representing each finite group

Frucht's theorem says every finite group is isomorphic to the automorphism group of a simple graph $G$ (with no loops, multiple edges or directed edges). There has been interest in finding classes of ...
21 votes
2 answers
1k views

Generating random finite groups

I would like a method to efficiently generate a random finite group of a given order $n$. If there are $g(n)$ non-isomorphic groups of order $n$, ideally each group would occur with probability $1/g(n)...
3 votes
1 answer
162 views

Does a finite non-abelian Group $G$ have a primitive nonabelian quotient $G/N$?

Let $G$ be a finite nonabelian group which is transitive with degree $d$. It is understood that we may construct a primitive group $H = P \wr G$ where $P$ is primitive such that $H/P^d = G$. Now I'm ...
9 votes
2 answers
167 views

Coboundary matrix of bar resolution for group cohomology: do the elementary divisors always divide $|G|$?

Consider the coboundary matrix $C^1(G, \mathbb{Z}) \to C^2(G, \mathbb{Z})$ of the normalized bar resolution of $G$ with coefficients in the trivial $\mathbb{Z}G$-module $\mathbb{Z}$. That is, thinking ...
11 votes
1 answer
331 views

A question on groups having a subgroup which fixes a vector in every irreducible representations

Given a finite group $G$, I am interested in finding a non-trivial proper subgroup $H$ of $G$ such that $\mathrm{Ind}_H^G\mathbf{1}$ contains all the irreducible representations of $G$, that is, ...
0 votes
1 answer
155 views

Combinatorial problem in $G(54, \, 5)$ - Reprise

This post is a follow-up to my previous post MO479127. I am trying to concentrate on a subset of relations, hoping to find some structure on the set of solutions that explains why the whole set of ...
3 votes
0 answers
359 views

Combinatorial problem in $G(54,\, 5)$

I have asked (probably) easier versions of this question in the past, see MO379272 and MO380292. At the moment, it is not clear to me how the beautiful answers to those questions can be helpful here. ...
12 votes
1 answer
654 views

Abelian group and $(\mathbb{Z}/3 \mathbb{Z}) ^k \rtimes \lbrace -1,1 \rbrace$

If $G$ is a finite group, let $A(G):=\forall a,b \in G, \exists n \in \mathbb{Z}, aba^{-1}b^{-1}=b^n a^n$ and $B(G):=\forall a,b \in G, \exists n \in \mathbb{Z}, aba^{-1}b^{-1}=b^n a^{-n}$. Is it true ...
37 votes
0 answers
1k views

Is this generalized character always a character?

Let $G$ be a finite group, and $p$ be a prime. Then there is a generalized character $\Psi$ of $G$ which takes value $0$ on all elements of order divisible by $p$, and has $\Psi(y)$ equal to the ...
8 votes
0 answers
190 views

Groups having exactly two non real-valued irreducible characters

This is an enlarged version of my question on MSE. It was suggested I ask here instead. Suppose the finite group $G$ has exactly two conjugacy classes that are not self-inverse (a conjugacy class is ...
13 votes
1 answer
370 views

Factorizing groups into a product of solvable subgroups

Does every finite group $G$ have a factorization $G=H_1\cdots H_k$ where the $H_i$ for $1\le i\le k$ are solvable subgroups of $G$ and $|G|=|H_1|\cdots |H_k|$ (equivalently, every element of $G$ is ...
13 votes
2 answers
2k views

Generalization of a theorem of Øystein Ore in group theory

Theorem (Øystein Ore, 1938): A finite group $G$ is cyclic iff its lattice of subgroups $\mathcal{L}(G)$ is distributive. Proof: see below. Let $(H \subset G)$ be an inclusion of finite groups and $\...
7 votes
0 answers
224 views

How many values of a group cocycle are required to know its cohomology class?

Suppose I have a finite group $G$, and a group cocycle $\varphi\in Z^n(G,U(1))$ (trivial action on $U(1)$) evaluated at $k$ distinct values of the inputs $g_1,...,g_n$. That is, I am given a set of ...
8 votes
1 answer
703 views

When does a finite group have finitely many indecomposable representations?

Let $G$ be a finite group and let $k=\mathbb F_p$. Then it is well-known that $G$ has finitely many irreducible modules. However, in general $G$ does not have finitely many indecomposable ...
1 vote
0 answers
85 views

inverse Galois problem on cyclic groups

It is known that the splitting field of $x^{p^n}-x$ over $\mathbb{F}_p$ is $\mathbf{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)\cong\mathbb{Z}/n\mathbb{Z}$ and the splitting field of $\Phi_n(x)$ over $\mathbb{...
1 vote
0 answers
127 views

Truchet tiles with non-periodic tiling from finite group multiplication tables (Thue-Morse plane)?

Let $G$ be a finite group with $n = |G|$ elements. By Cayley's theorem for finite groups, we have an injective homomorphism of groups: $$ \pi : G \rightarrow S_n, \quad g \mapsto \pi(g) $$ where ...
4 votes
1 answer
421 views

Visualizing the elements of a finite group and does the Gram matrix determine the finite group?

Let $G$ be a finite group with $n = |G|$ elements. By Cayley's theorem for finite groups, we have an injective homomorphism of groups: $$ \pi : G \rightarrow S_n, \quad g \mapsto \pi(g) $$ where ...
1 vote
0 answers
109 views

Center of factors of a finite $p$-group, obtained from a minimal normal subgroup

throughout a research problem about finite $p$-groups, I have a challenge as follows, Let $G$ be a finite non-abelian $p$-group, where $p$ is odd and $Z(G)$ is non-cyclic. ($Z(G)$ denotes the center ...
1 vote
0 answers
92 views

Finite groups whose center nontrivially represented in irreps with coprime dimensions

I have been searching for a finite non-abelian group $G$ with the following properties: Its center $Z(G)$ acts as the identity in all dimension one irreps (i.e., $Z(G)$ is a subgroup of the ...
1 vote
0 answers
72 views

Normalizer of connected subgroup contained in the Weyl group?

Let $ G $ be a simple Lie group. Let $ H $ be a connected subgroup of $ G $ such that $ N(H)/H $ is finite. In such a case, is $ N(H)/H $ always a subgroup of the Weyl group of $ G $? For $ G=\...
2 votes
0 answers
121 views

A-conjugately dense subgroup

A subgroup $H$ of a group $G$ is called conjugately dense in $G$ if $H$ has nonempty intersection with each class of conjugate elements in $G$. We know that if $G$ is finite, then $H=G$. Now, my ...
5 votes
1 answer
211 views

The rank of indecomposable finite abelian 2-group

$\DeclareMathOperator\rank{rank}$Let $P$ be a finite $p$-group. The rank of $P$ is $\log_{p}|P/\Phi(P)|$ where $\Phi(P)$ is the Frattini subgroup of $P$, we write $\rank(P)=\log_{p}|P/\Phi(P)|$. Let a ...
4 votes
2 answers
313 views

Structure of Sylow $p$-subgroup of $G$ with given property

Let $G$ be a finite group and $A\leq \operatorname{Aut}(G)$. Assume that $‎P$ is a Sylow $p$-subgroup of $G$ and $z\in Z(P)$ of order $p$ such that each ‎subgroup of order $p$ of $P$ is $A$-conjugate ...
3 votes
1 answer
149 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
318 views

Can $\text{Aut}(G)$ be extended to contain $G$?

Let $G$ be a group (finite, say) with center $Z$. The automorphism group $\text{Aut}(G)$ sits in a short exact sequence $$ 1 \to G/Z \to \text{Aut}(G) \to \text{Out}(G) \to 1. $$ So when $Z\neq 1$, as ...
2 votes
1 answer
345 views

A natural automorphism of a finite group with two generators?

I am looking for a proof or a counterexample for the following: Let $G$ be a finite group generated by $f$ and $g$. The map $f\mapsto f^{-1}, g\mapsto g^{-1}$ can be extended to an automorphism $\...
9 votes
2 answers
794 views

Why do these finite group Dedekind matrices seem to have integer spectrum when specialized to the order of group elements?

Let $p$ be a prime and let $f_p$ be the permutation on the set $\{1,2,\cdots,p-1\}$ which is given by taking inverses in $\mathbb{Z}/(p)$: $$x \bmod(p) \mapsto \frac{1}{x} \bmod (p)$$ So for instance, ...

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