# Questions tagged [gr.group-theory]

Questions about the branch of algebra that deals with groups.

7,102
questions

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### Size of subsets of nilpotent groups

Let $\Gamma$ be a nilpotent group finitely generated by a set $S$
with $S=S^{-1}$ and write $[\Gamma,\Gamma]$ for the commutator subgroup.
The abelianization $\Gamma/[\Gamma,\Gamma]$ is a finitely ...

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1
answer

76
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### Perfect group that is split extension of a normal free subgroup of finite index

Does there exists a non-trivial free group $F$ and a finite group $L$ acting on $F$ such that the semidirect product $F\rtimes L$ is perfect?
Thanks @YCor for reformulating the question.

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97
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### Isomorphism of invariants and coinvariants over a field

Let $G$ be a finite group with normal subgroup $N$ acting on a vector space $V$ over a field $k$ in which the order of $N$ is invertible. Denote $H:=G/N$. The composite map $V^N \to V \to V_N$ and $\...

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### $S$-semipermutable subgroup

A subgroup $H$ of a finite group $G$ is said to be
$S$-semipermutable in $G$ if it permutes with every Sylow $q$-subgroup of $G$ for all primes $q$ not dividing $|H|$.
Assume that $G$ is solvable and ...

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### What are the indecomposable modules over $\mathbb{F}_2(C_2\times C_2)$?

Let $C_2$ be the cyclic group of order $2$ and $\mathbb{F}_2$ the field with $2$ elements. Consider the group algebra $A:= \mathbb{F}_2 (C_2\times C_2)$. It is well-known that $A$ has infinite ...

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2
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### What is the quotient group $D^*/{F^*(1+P_D)}$ for a quaternion division algebra $D$ over a local field $F$?

Let $F$ be the non-archimedean local field $\mathbb{Q}_p$ for some prime $p$ and $D$ be a quaternion division algebra over $F$. Let $\mathcal{O}_D$ and $\mathcal{P}_D$ denote the ring of integers of $...

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107
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### Conjugacy classes of $PSL_2(11)$ and $PGL_2(11)$ in $Aut(HN)$

How many conjugacy classes each of $PSL_2(11)$ and $PGL_2(11)$ subgroups are contained in the automorphism group of the Harada-Norton group?

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119
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### Finiteness of $\ell^2$-Betti numbers

I'm reading the paper "Improved algebraic fibering" by Sam Fisher (https://arxiv.org/pdf/2112.00397.pdf) and in the proof of lemma 6.4 it claims the followng:
$(\mathcal{D}_{\mathbb{F}K}\ast\...

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108
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### Distortion in the Brin-Thompson 2V

Is it known whether the Brin-Thompson 2V contains a distortion element? By this I mean an element $f$ such that the word norm $|f^n|$
grows sublinearly, and $f$ is of infinite order. If such an ...

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### Uncountable locally finite group contains a countably infinite normal subgroup

Is it true that every uncountable locally finite group contains a countably infinite normal subgroup? If not, is there a counter example of an uncountable locally finite group that has no countably ...

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### Known non-solvable inseparable subgroups of the Monster

I’m looking for a list of the known isomorphism classes of non-solvable inseparable subgroups of the Monster sporadic group, along with the number of conjugacy classes for each that it is known.

2
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### How far is a countably infinite reduced abelian $p$-group from being an infinite direct sum?

Question Let $G$ be a countably infinite reduced abelian $p$-group. Is it always possible to write it has an infinite direct sums of non-trivial groups? If it is not true, how far is $G$ from being an ...

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1
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### Element of order $p$ and finite height $\geq1$ in a reduced abelian group $p$-group with an element of order $p^2$

This is a reference request for the following statement:
Fact:
Let $G$ be a reduced abelian $p$-group with an element of order $p^2$. Then $G$ contains an element of order $p$ and of finite height at ...

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### Are the non-free factors of Grushko decomposition of a finitely generated convex-cocompact (but not cocompact) subgroup of PSL$(2,\mathbb{R})$ finite?

See Grushko decomposition theorem.
Are the non-free factors of Grushko decomposition of a finitely generated convex–cocompact (but not cocompact) subgroup of $\operatorname{PSL}(2,\mathbb{R})$ finite?
...

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votes

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### Extensions containing the Schur cover II

Given two finite groups $G$ and $H$ such that $H$ is a perfect subgroup of $G$, does there always exist a finite solvable group $I$ such that the Schur cover of $H$ embeds into an extension of shape $...

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163
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### The center of Sylow subgroups

$\DeclareMathOperator\Syl{Syl}$Let $G$ be a finite group, $\Phi(G)$ is the Frattini subgroup of $G$. And $G/\Phi(G)$ is a simple group. Let $P\in \Syl_{p}(\Phi(G))$, where $p\in \pi(\Phi(G))$ and $\...

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94
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### Hamilton decomposition for infinite Cayley graphs

We have a finitely generated group $G$ which is infinite. Does there exist such a finite generating set $S$, $G= \langle S\rangle$, that the corresponding Cayley graph $\mathrm{Cayley}(G,S)$ can be ...

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### God's number for higher dimensional Rubik's cubes

In this MO question, user Martin Brandenburg asks about God's number for $n \times n \times n$-cubes for $n>3$. Here, God's number $g(n)$ was defined as the smallest number $m$ such that every ...

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### Finite index subgroup of $\mathbb{Z}^2$ that is invariant under a non-singular matrix

Let $M $ be a matrix in $ \operatorname{GL}(2, \mathbb{Z})$ that has at least one eigenvalue of absolute value strictly bigger than $1$. What are the finite index subgroups $H$ of $\mathbb{Z}^2$ such ...

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268
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### What about a Cayley n-complex for n>2?

Let $G$ be a finitely presented group. The Cayley graph of the finite generating set is a $1$-complex where the $0$-cells are the elements of $G$ and the $1$-cells are given by the generators (...

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### Presentation complex of a finite perfect group and its features

Let $G$ be a finite perfect group and consider $X_G$, its presentation complex. I have the following questions:
Is there any special property of $X_G$ due to the group's perfectness?
What can we say ...

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### What is mean of action of group on tower?

What is mean of action of group on tower ?in the page 17 of following paper (action on teichmuller tower )
In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class ...

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84
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### Are there standard short notations for ascending and descending cyclic permutations?

In a paper I am currently writing I use cyclic permutations of the form
$$
(k,k+1,\dots,\ell)
$$
and
$$
(\ell,\ell-1,\dots,k)
$$
of consecutive elements quite a lot (I added the commas to avoid ...

3
votes

1
answer

153
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### Extensions containing the Schur cover

Given two finite groups $G$ and $H$ such that $H$ is a perfect subgroup of $G$, is there always an extension of a finite group by $G$ such that the image of $H$ under the extension is isomorphic to ...

3
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0
answers

73
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### Terminology for the "natural probability measure" on the set of irreducible characters of a finite group

To be specific: if $G$ is a finite group and $\operatorname{Irr}(G)$ the set of its irreducible characters (over the complex field) then we know that
$$
1 = \sum_{\phi \in {\rm Irr}(G)} \frac{(d_\phi)^...

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votes

2
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520
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### Group of exponential growth always contains a free sub-group?

I am not very conversant with the growth of a group, so this may be a very silly question.
It is known that $F_2$, the free group of rank $2$, has exponential growth. I was wondering whether the ...

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### Minimal non-solvable groups with a special property

Suppose that $G$ is a finite minimal non-solvable group (i.e. a non-solvable group
all of whose proper subgroups are solvable) whose Frattini subgroup $\Phi(G)$
is abelian. If for every subgroup $H &...

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413
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### New (?) math object. Looking for (if existing) literature [closed]

I am interested in any literature about the following mathematical property.
Let $V$ a vector space and $G$ a group acting on $V$.
What is the name for the property of a set of operators $H=\{h:V\to V\...

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176
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### Non-abelian group cohomology, additional information

Let $G$ be a (profinite) group, and let $M$ be a non-abelian $G$-module.
We know how to construct reasonably $H^0(G,M)$ and $H^1(G,M)$ and it turns out that $H^1(G,M)$ is just a pointed set and not ...

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### What is "inn" in this paper?

In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. 521, 1-24 (2000). ZBL0953.20030.,in the page 18 we have:
$$
\begin{aligned}
...

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### Does a basic commutator of the cartesian subgroup belong to the second subgroup of the derived series?

Let $G$ and $H$ be groups and consider $\pi: G\ast H \to G\times H$ the canonical morphism. The cartesian subgroup is defined as $[G,H]:=\ker(\pi)$. I will call $s_2(G\ast H)$ the second subgroup of ...

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### Algorithmically handling the Spin groups in larg(ish) dimensions

Question: Is there a reasonably efficient algorithmic representation of $\mathit{Spin}_n$? By this I mean, a way to store its elements and operate on them (multiply, inverse, maybe compute ...

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348
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### Is every virtually free group residually finite?

Question: Is every (finitely generated) virtually free group residually finite?
A well-known question asks whether every hyperbolic group is residually finite (Mladen Bestvina. Questions in geometric ...

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votes

1
answer

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### Is there a classification of homomorphisms $S_n \to S_{n+k}$ for small $k$?

Homomorphisms $B_n \to B_{2n}$ and $B_n \to S_{2n}$ have been classified in Chen–Kordek–Margalit - Homomorphisms between braid groups and Lin - Braids and permutations respectively. I am interested in ...

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### combinatorial way to count representatives of conjugacy class of elements of ord 5

I am trying to find a representative of each conjugacy class of order 5 elements in PGL$_6$($\mathbb C$).
Let $r$ in $\mathbb C$ such that $r^5 = 1$ and [ ] denote modular the center of GL$_6(\mathbb ...

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### Classification of elements $GL(d, \mathbb{R})$

Any $SL(2, \mathbb{R})$ is either elliptic or hyperbolic, or parabolic up to conjugacy; see here.
Do we have the same classification for $GL(d, \mathbb{R})$? If so, could you please introduce some ...

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2
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167
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### Reflection quotients of Coxeter groups

I am interested in a concept somehow dual to reflection subgroups. A reflection quotient of a Coxeter system $(W, S)$ shall be a surjective homomorphism $W \to W'$ to a Coxeter group $W'$ such that ...

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### Classification of finite minimal non-solvable groups

Is there any classification of finite minimal non-solvable groups? By minimal non-solvable group, I mean a non-solvable group all of whose proper subgroups are solvable.

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### Row of the character table of symmetric group with most negative entries

The row of the character table of $S_n$ corresponding to the trivial representation has all entries positive, and by orthogonality clearly it is the only one like this.
Is it true that for $n\gg 0$, ...

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1
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### Is this semi-direct product residually finite?

I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts here can help me find a way to check the residual finiteness of this group.
Consider the ...

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### Infinite vertex-transitive graph where every automorphism has a fixed vertex

This is a follow-up to the question Connected vertex-transitive graph with the fixed-point property. In particular, it is based on a comment by user bof.
Let $G = (V,E)$ be a graph with $V$ infinite. ...

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107
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### The intersection of product of Sylow subgroups

Suppose $G$ is solvable, and $\pi(G)= \{2,m,n\}$, $O_{2}(G)=1$. Then can we use the solvability of $G$ to prove that $O_{2^{\prime}}(G) \neq 1$? Let $\bar{G}= G / O_{2^{\prime}}(G)$, what about $O_{...

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### From direct sum of quotient group of a group to direct sum of the group

We know that for a cyclic group $G$, if $G=A\oplus B$, then for some subgroups $H$ of $G$, We have $G/H=(A+H)/H\oplus (B+H)/H.$ But, if we know that for a subgroup $H$ of $G$, $G/H=(A+H)/H\oplus (B+H)...

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### Can we define partial group actions on (finite) sets via generators and relators?

Let $G = \langle Y | R \rangle$ be a finitely presented group. A partial group action on a set $X$ is a premorphism into the inverse semigroup
$$
\mathcal I (X) = \{ f: A \to B : A, B \subseteq X, f\...

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### A particular permutation on $\mathbb{Z}_n$

Let us consider $\mathbb{Z}_N=\{0,1,\cdots N-1\}$.
Does there exist any permutation $\sigma$ on $\mathbb{Z}_N$ satisfying :
$$\exists p,q\in\mathbb{N}, ~~p,q\neq0 (mod~N) ~~\forall i,j\in \mathbb{Z}...

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### Different definitions of p-fusion and Mislin's theorem

Currently, I am trying to understand and compute homology of finite groups with coefficients in a field of positive characteristic. So, I was searching for some results that could reduce this problem (...

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1
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### Group theory with grep?

While reading Bill Thurston's obituary in the Notices of the AMS I came across the following fascinating anecdote (pg. 32):
Bill’s enthusiasm during the early stages of mathematical discovery was ...

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1
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71
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### Algorithm to find a minimal normal subgroup of given group $G$ by matrix group representation

Given a matrix group $G$ by its generators i.e. $G =\langle A_1,A_2,...,A_k \rangle \leq GL_n(q)$, where each $A_i$'s are matrix in $GL_n(q)$
Q. Does there exist a polynomial time (polynomial in ...

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### Characters of upper triangular matrices over finite field - reference request

Let $B_n$ be the group of upper matrices and $U_n$ the subgroup of unipotent upper triangular matrices. I would like some references which discusses complex character theory of $B_n(\mathbb{F}_q)$ for ...

6
votes

1
answer

358
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### Does a non-simple perfect group always have a maximal subgroup whose derived subgroup has nontrivial core?

Let $G \neq 1$ be a finite perfect group which is not simple.
Is it true that $G$ necessarily has a maximal subgroup whose derived subgroup
has nontrivial core in $G$?
Remark 1: This holds for all ...