Questions tagged [finite-groups]
Questions on group theory which concern finite groups.
143 questions from the last 365 days
2
votes
0
answers
60
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 ...
- 71
1
vote
1
answer
260
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 $...
- 171
12
votes
0
answers
339
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 $\operatorname{PGL}_d(R)$?
...
- 602
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 $g$, ...
- 1,354
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 ...
- 4,547
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 ...
- 1,150
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 \...
- 361
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 ...
- 1,856
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}...
- 217
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: ...
- 24.9k
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 ...
- 24.9k
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?
...
- 393
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 $\mathbb{F}_q$...
- 1,150
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
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 $p_i$ ...
- 24.9k
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
$$ G = \bigsqcup_{b\in B} bA.$$
...
- 1,354
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 ...
- 1,150
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 ...
- 605
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 ...
- 361
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 ...
- 231
10
votes
0
answers
423
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 ...
- 339
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 ...
- 1,926
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 ...
- 3,230
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 ...
- 66.3k
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, ...
- 251
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 ...
- 353
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 ...
- 787
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.
...
- 66.3k
8
votes
1
answer
702
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 ...
- 3,054
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{...
- 11
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 ...
- 3,054
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 ...
- 219
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 ...
- 44.4k
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 ...
- 11
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=\...
- 4,081
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 ...
- 111
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 ...
- 787
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 ...
- 173
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 ...
- 3,054
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 ...
- 173
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 ...
- 815
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 $\...
- 131
7
votes
1
answer
282
views
Galois groups of truncated $\cosh(x)$ Taylor polynomials and related results?
(Part of this question was written with ChatGPT because english is not my native language).
I am currently translating my diploma thesis from 2010 in english and thought to think about the topic again:...
- 3,054
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, ...
- 3,054
5
votes
1
answer
258
views
Finite simple $\pi$-groups
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$Let $\pi$ be a finite set of primes. A finite group $G$ is a $\pi$-group if all primes dividing $|G|$ lie in $\pi$. Is it true that there ...
- 56.9k
2
votes
0
answers
97
views
Centralizer bound for irreducible representations of $\operatorname{SU}_n(\mathbb{C})$
Let $G$ be a finite group, $χ$ be the character of an irreducible representation $V$ of $G$, and $g ∈ G$. Then a classical bound on the trace of $g$ is given by: $|χ(g)|² ≤ |C_G(g)|$, where $C_G(g)$ ...
- 2,907
13
votes
1
answer
420
views
Embedding rank of finite groups and quotients
Let $G$ be a finite group, and $n$ a positive integer. It is not hard to check that the following are equivalent:
For every $g\in G\setminus\{1\}$ there is a subgroup $H\leq G$ with $|G/H|\leq n$ ...
- 56.9k
3
votes
0
answers
87
views
Stem extensions and quotients of Schur covers
Suppose that $G$ is a finite group, and that $\Gamma$ is a central extension of $G$ by $A$, that is
$$ 1 \rightarrow A \rightarrow \Gamma \rightarrow G \rightarrow 1$$
with the image of $A$ contained ...
- 1,300
7
votes
1
answer
328
views
A projectivity property in the category of groups
Let $F_r$ be the free group on $r$ generators, let $G$ and $H$ be finite groups, and let $F_r\xrightarrow{\alpha}H\xleftarrow{\beta}G$ be surjective homomorphisms. It is then easy to see that we can ...
- 56.9k