All Questions
Tagged with finite-groups gr.group-theory
1,677 questions
2
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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
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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
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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
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0
answers
163
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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
2
votes
2
answers
206
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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
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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
2
votes
0
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100
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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
8
votes
1
answer
1k
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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
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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
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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
115
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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
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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
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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
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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
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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
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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
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0
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423
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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
9
votes
2
answers
167
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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
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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
12
votes
1
answer
654
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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
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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
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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
37
votes
0
answers
1k
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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
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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
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0
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109
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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
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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
2
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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
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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
5
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1
answer
258
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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
13
votes
1
answer
420
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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
1
vote
0
answers
125
views
Generators of a Coxeter group
Let $(W,S)$ be a Coxeter system and assume that $|S|$ is finite. Certainly, $W$ is generated by $|S|$ simple reflections. My question is: Can $W$ be generated by fewer reflections? (Including non-...
- 71
14
votes
2
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725
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Are there any non-conjugation "extendible automorphisms" in the category of finite groups?
Let $\mathbf{Grp}$ be the category of groups. Given a subcategory $\mathscr{G}$ of $\mathbf{Grp}$ and $G\in\mathit{Ob}(\mathscr{G})$, a $\mathscr{G}$-extendible map on $G$ will here mean an assignment ...
- 21.2k
1
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0
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172
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Isomorphism classes of finite $\mathbb{N}$-groups
Where can I find resources on isomorphism classes of finite $\mathbb{N}$-groups, i.e. groups acted on by the monoid $(\mathbb{N}, +)$?
I edited this question to be more focused on what I'm interested ...
- 591
11
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1
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330
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What is the minimal genus of a surface acted on by the symmetric group $S_n$?
For $G$ a finite group, it is easy to construct a (connected, orientable) surface with a faithful action of $G$. E.g.: take a disjoint union of $G$ many spheres, and add a 1-handle for every edge in ...
- 43.2k
2
votes
0
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99
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Cohomological characterization of when $f: \pi_1(\Sigma_g) \to P$ factors through $F_g$ when $P$ is perfect
In previous questions on this site such as this one, it has been asked when a map $\varphi \colon G \to H$ of finitely generated groups factors through a free quotient meaning that there exists a ...
- 71
2
votes
1
answer
273
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Equivariant Smith normal form?
Let $F$ be a finitely generated free $\mathbb{Z}$-module on which the group $G$ of two elements acts via group homomorphisms. Let $F'$ be a $G$-invariant submodule. By Smith normal form we know that ...
- 3,031
1
vote
0
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103
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Normality of the intersection between a Carter subgroup and the nilpotent residual of a solvable group G
In his book "Group Theory," Schenkman, in the proof of Theorem VII.4.a, states that in a finite solvable group $G$, the intersection of a Carter subgroup $C$ (i.e., a self-normalizing and ...
8
votes
1
answer
353
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Structure of a single automorphism of a finite abelian p-group
A finite abelian $p$-group $H$ is homogenous when it is the direct sum of cyclic groups of the same order $p^r$, i.e. $H \cong \big(\mathbb{Z}/p^{r}\mathbb{Z}\big)^{e}$. Every finite abelian $p$-group ...
- 365
2
votes
0
answers
153
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How good is approximation of distance function on the Cayley graph by "Fourier" basis coming from the irreducible representations?
Consider finite group $G$ , symmetric set of its elements $S$, construct a Cayley graph.
Consider $d(g)$ - word metric or distance on the Cayley graph from identity to $g$.
As any function on a group ...
- 24.9k
2
votes
1
answer
111
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Structure of elements of a finite group not contained in any conjugate of a proper subgroup
Let $G$ be a finite group and $H$ be a proper subgroup of $G$. It is elementary to prove that the union of all conjugates of $H$ under $G$,
$$U:=\bigcup_{\sigma\in G}\sigma^{-1}H\sigma,$$
is properly ...
3
votes
1
answer
182
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Schur cover of alternating groups
Wilson's book "The finite simple groups" gives (in section 2.7) a description of the double cover of the alternating groups. First, one constructs a double cover $2S_n$ of the symmetric ...
- 7,527
7
votes
0
answers
145
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Transitive groups with fixed-point free elements of prime power order
A well-known result of Fein, Kantor and Schacher says that if $G$ is a finite group which acts transitively on a set $X$, then $G$ contains an element of prime power order without fixed letters. ...
- 4,547
9
votes
1
answer
335
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Finite p-group $G$ of exponent $>p$ with all elements outside $\Phi(G)$ of order $p$
Does there exist a finite $p$-group $G$ of exponent $>p$, such that $o(g)=p$ for all $g\in G\setminus\Phi(G)$?
- 171
9
votes
2
answers
438
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Irreducible tensor product representations in finite simple groups
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\PSp{PSp}\DeclareMathOperator\PSU{PSU}$Background:
A representation $ \rho: G \to \GL(V) $ of a group $ G $ on a (complex) ...
3
votes
1
answer
222
views
A question about coprime automorphisms of profinite groups
Let $p$ a prime. A finite group is a $p'$-group if its order is prime to $p$. Let $A$ be a finite $p'$-group of automorphisms of a finite $p$-group $G$. Suppose that $A$ is a non-cyclic abelian group. ...
- 863
12
votes
2
answers
698
views
Generators of a group and normal subgroups
Can we say anything about a minimal generating set of a finite group based on its normal subgroups? For example, can we bound their order, or say whether they come from the same conjugacy class?
An ...
- 175
0
votes
1
answer
205
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Hyperoctahedral group, preliminaries [closed]
I am looking for information on the hyperoctahedral group
From what I understand, the hyperoctahedral group is 'the generalized symmetric group' in the case where $m=2$. That is, the hyperoctahedral ...
- 111