# Questions tagged [gr.group-theory]

Questions about the branch of algebra that deals with groups.

6,634
questions

**3**

votes

**1**answer

187 views

### Is there a higher categorical structure which models the (higher) conjugation actions of a group acting on itself?

Let $G$ be a group, and consider the action of $G$ on itself by conjugation. If we think of $G$ as a one object category, then the conjugation action can be realised as automorphisms of this category, ...

**0**

votes

**0**answers

78 views

### Large subgroups of infinite-dimensional vector spaces

Let $V$ be an infinite-dimensional vector space over $\mathbb{Q}$.
Consider a proper subgroup $W$ of $V, +$ with the following property: each vector line $L$ (which we see as a subgroup of $V, +$) has ...

**4**

votes

**4**answers

406 views

### The integral closure $\overline{\mathbb{Z}}$ and the group $\overline{\mathbb{Z}}^{\times}$

Let $\mathbb{Q}$ be the field of rational numbers, and let $\overline{\mathbb{Q}}$ be its algebraic closure. Assume $\overline{\mathbb{Z}}$ is the integral closure of $\mathbb{Z}$ in $\overline{\...

**9**

votes

**1**answer

136 views

### Group extensions with non-abelian kernel

If $N$ is a normal subgroup of $G$ then there is a coupling: that is, a representation of $G/N$ in $\operatorname{Out}(N)$. In that case, the extensions of $N$ by $G/N$ affording the same coupling are ...

**2**

votes

**0**answers

50 views

### Restrictions on pointed lifts of isometries

Let $M$ be a closed Riemannian manifold and let $f$ be an isometry of $M$ that fixes a point $\ast \in M$ and acts trivially on $\Gamma := \pi_1(M,\ast)$. Then there is a unique isometry $\tilde{f}$ ...

**2**

votes

**1**answer

117 views

### A variation of closed-subgroup theorem

$\DeclareMathOperator\SO{SO}$Recall that the closed-subgroup theorem (Wikipedia link) says that a closed subgroup of a Lie group is a Lie group.
I am pretty sure that this theorem should have a "...

**1**

vote

**0**answers

35 views

### Ideal Ford domain (for finite index subgroup)

Let $G$ be a lattice Fuchsian group with parabolic elements, seen as a discrete subgroup of matrices
$
g=
\begin{pmatrix}
\alpha & \overline{\beta} \\
\beta & \overline{\alpha}
\end{pmatrix}
$...

**6**

votes

**1**answer

171 views

### Is the function $k(g,h) = \frac{1}{1+\lvert gh^{-1}\rvert}$ positive definite?

Let $G$ be a finite group, $S \subset G$ a generating set, closed under taking inverses, and $\lvert\cdot\rvert$ the word length with respect to this set $S$.
Question. Is the function $k(g,h) = \...

**3**

votes

**0**answers

82 views

### Ping pong with parabolic isometries on Gromov hyperbolic spaces

For a group $G$ with a non-elementary general type action by isometries on a Gromov hyperbolic geodesic space $(X,d)$, it is well known that you can construct free subgroups of $G$ via the ping pong ...

**1**

vote

**1**answer

116 views

### Subgroup rank of finite simple groups

Definition: The subgroup rank of a finite group $G$ is the minimal natural number $n$ such that every subgroup of $G$ can be generated by $n$ elements (or fewer).
This invariant has been studied ...

**2**

votes

**0**answers

59 views

### Weights of finite abelian group actions on submanifolds/subvarieties

(cross-posted from https://math.stackexchange.com/questions/4125529/weights-of-finite-abelian-group-actions-on-submanifolds-subvarieties)
How do weights associated to actions of finite subgroups of $\...

**3**

votes

**1**answer

68 views

### Example of a primitive group of affine type and of twisted wreath product type

Good evening,
According to the O'Nan-Scott theorem, primitive finite group are classified into five classes: affine type, product type, almost simple type, diagonal type and twisted wreath product ...

**8**

votes

**1**answer

143 views

### On the coefficients that appear in finite groups of matrices with integer entries

Let $n$ be a positive integer and $G$ be a finite group of $n\times n$ matrices with integer coefficients, i.e. $G\subset\operatorname{GL}_n(\mathbb{Z})$. It is known that for sufficiently large $n$, ...

**-4**

votes

**1**answer

282 views

### Are there overwhelmingly more finite monoids than finite spaces? [closed]

A function $f:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ overwhelms $g:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ if for any $k\in \mathbb{Z}_{\geq 1}$ the inequality $f(n)\leq g(n+k)$ holds only for ...

**3**

votes

**1**answer

371 views

### Are there overwhelmingly more finite posets than finite groups? [closed]

A function $f:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ overwhelms $g:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ if for any $k\in \mathbb{Z}_{\geq 1}$ the inequality $f(n)\leq g(n+k)$ holds only for ...

**4**

votes

**0**answers

143 views

### Terminology question in group actions

Given a continuous group action $G \times X \rightarrow X$ on a topological space $X$, is there a standard term for the subsets $K \subset X$ for which
Every open neighborhood of $K$ intersects every ...

**9**

votes

**0**answers

164 views

### Hopficity of Baumslag-Solitar groups

I am struggling to find the exact source (with proofs) of the following ''well-known'' statement:
the Baumslag-Solitar group $BS(m,n)=\langle a,t \mid ta^m t^{-1}=a^n\rangle$ is Hopfian if and only if ...

**7**

votes

**1**answer

195 views

### Fusing conjugacy classes

Consider a finite group $G$ and two conjugacy classes $H$ and $I$ of isomorphic subgroups of $G$.
Question. Is there some finite overgroup of $G$ which fuses $H$ and $I$ into a single conjugacy class?...

**7**

votes

**0**answers

98 views

### Can a lacunary hyperbolic group be Liouville?

While discussing with a colleague of a possible use of lacunary hyperbolic elementarily amenable groups introduced by Olshanskii, Osin & Sapir, it occurred to me that I was not aware whether ...

**7**

votes

**2**answers

218 views

### Index of the mapping class group $\Gamma_{g,n}$ inside $\text{Out}(\Pi_{g,n})$

Let $\Sigma_{g,n}$ denote an $n$-punctured surface of genus $g$, with $2g+n-2 > 0$. Let $\Pi_{g,n}$ be its fundamental group (for some choice of base point), and let $\Gamma_{g,n}$ denote its pure, ...

**3**

votes

**0**answers

95 views

### Torsion-free subgroups in arbitrary lattices

Let $\Gamma $ be a lattice in a semi-simple Lie group $G$. If the Lie group $G$ is linear (that is, it has a faithful finite-dimensional linear representation), then $\Gamma$ contains a torsion-free ...

**8**

votes

**1**answer

374 views

### Definition of an arithmetic subgroup of an algebraic group

I'm struggling with the definition of an arithmetic subgroup of an algebraic group defined over $\mathbb{Q}$.
In Wikipedia you can read:
If $\mathrm G$ is an algebraic subgroup of $\mathrm{GL}_n(\...

**7**

votes

**1**answer

305 views

### Independent vectors in the permuting coordinates action of $S_n$ on $\mathbb{R}^n$

Let $V$ be the hyperplane in $\mathbb{R}^n$ with equation $\sum_i x_i=0$. The symmetric group $S_n$ acts on $V$ by $s\cdot (v_1,\ldots,v_n)=(v_{s^{-1}(1)},\ldots,v_{s^{-1}(n)})$. Consider those $v\in ...

**4**

votes

**1**answer

89 views

### Computable change in minimum word length of subgroup elements

Let $G$ be an infinite finitely generated group. Fix a finite generating set for $G$.
Define $\mathrm{len}_G:G\to\mathbb{Z}_{\geq 0}$ by sending $g$ to the minimum length of a word in the generators ...

**7**

votes

**1**answer

199 views

### Are two quasi-isometric, isomorphic on large enough balls, transitive graphs isomorphic?

Take two transitive graphs $X,Y$ (potentially directed and edge-labelled, e.g. Cayley graphs).
Assume $X,Y$ are quasi-isometric with constant $K$, i.e. there exists a function $f:VX \to VY$ ($VX,\,VY$ ...

**3**

votes

**0**answers

90 views

### Finitely presented group with exactly two conjugacy classes

Is there a finitely presented group with exactly two conjugacy classes other than $\mathbb{Z}/2\mathbb{Z}$?

**6**

votes

**0**answers

61 views

### A criterion for loxodromicity in Gromov-hyperbolic spaces

Recall that an isometry of a Gromov-hyperbolic space $X$ is called loxodromic if it has exactly two fixed points on the Gromov boundary $\partial X$, one being "attracting" and the other &...

**9**

votes

**0**answers

98 views

### Decidable membership for subgroup generated by three elements in $F_2\times F_2$

Let $F_2$ be the non-abelian free group on two generators. Let $G\subset F_2\times F_2$ be a subgroup generated by three elements. Is there an algorithm deciding if a given element of $F_2\times F_2$ ...

**7**

votes

**1**answer

113 views

### Infinite oscillation of minimum word length in 2-generated group

Let $G$ be a group with generators $a, b\in G$.
Define $\mathrm{len}:G\to\mathbb{Z}_{\geq 0}$ by sending $g$ to the minimum length of a word in $a, b, a^{-1}, b^{-1}$ equal to $g$.
Assume that for all ...

**2**

votes

**1**answer

270 views

### Is the representation of finite simple groups fully understood?

Is the representation of finite simple groups fully understood? To clarify, I mean have all the simple representations (even finite dimensional) been classified in terms of some classifying set, such ...

**0**

votes

**0**answers

89 views

### How to classify rings by combinatorial structures?

There are many ways to encode information about algebraic structures such as groups, rings, etc... in combinatorial form. For example the Cayley graph of a group with a subset of generators, or the ...

**7**

votes

**3**answers

351 views

### Membership to double cosets in free groups

Is there an elementary and efficient algorithm for testing the membership to a double coset of f.g. subgroups in a free group?
Has this membership problem been implemented in GAP/Magma?
More ...

**2**

votes

**0**answers

61 views

### Terminology and notation for generated subgroups

I would like to think about formation of the smallest subgroup (or monoid, or whatever) $H$ of $G$ containing two given subgroups $A$ and $B$ as an operation on subgroups, and I wonder if there is a ...

**7**

votes

**1**answer

230 views

### Induction and restriction of unitary representations

$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\Ind{Ind}\DeclareMathOperator\Res{Res}$Given a locally compact group $G$ and a closed subgroup $H\subset G$,
let $\Rep(G)$ and $\Rep(H)$ denote their ...

**6**

votes

**0**answers

128 views

### Is it possible for a finitely generated group to have polynomial growth rate with leading coefficient less than 1?

I'm quite a neophyte in this area, so apologies if this question is easy. (I've edited slightly to make my question more clear.)
I've been trying to learn about growth rates for finitely generated ...

**5**

votes

**1**answer

64 views

### Equality to a power of a given word undecidable in finitely presented group with decidable word problem

Let $G$ be a group with an explicit finite presentation. Assume $G$ has a decidable word problem.
Can there exist an explicit word $w\in G$ such that there is no algorithm deciding if a given word $w'\...

**6**

votes

**1**answer

111 views

### Empty preimage under homomorphism of finitely presented groups with decidable word problems

Let $f:G\to H$ be a homomorphism of finitely presented groups with decidable word problems.
Assume you are given explicit finite presentations for both $G$ and $H$ and you are given the words to which ...

**13**

votes

**1**answer

408 views

### Is the diagonal of finitely presented groups computable?

Let $f:G\to H$ be a surjective homomorphism of finitely presented groups. If the kernel of $f$ is finitely generated then is $G\times_H G$ is a finitely presented group? Can one compute an explicit ...

**16**

votes

**2**answers

794 views

### Element being trivial in a finitely presented group independent of ZFC

Is there an explicit finitely presented group $G$ and an element $g\in G$ such that the statement "$g$ is equal to the identity" is independent of ZFC?

**6**

votes

**1**answer

129 views

### Empty preimage under homomorphism of finitely presented groups independent of ZFC

Is there a homomorphism of finitely presented groups $f:G\to H$ and an element $h\in H$ such that the statement "$f^{-1}(h)$ is empty" is independent of ZFC?

**3**

votes

**0**answers

62 views

### Empty preimage under homomorphism of finitely presented groups with decidable word problems

Let $G, H$ be finitely presented groups with decidable word problems.
Can there be a homomorphism $f:G\to H$ such that there is no algorithm deciding given $w\in H$ whether $f^{-1}(w)$ is empty or not?...

**5**

votes

**0**answers

121 views

### Does any subset of a finitely generated group with positive upper density contain three points in arithmetic progression?

Let $G$ be a countable finitely generated group, with word metric $d_S$ induced by some generating set $S$. Let $B_r$ be the ball of radius $r$ around the identity under $d_S$.
We say that $A \subset ...

**2**

votes

**0**answers

45 views

### Quasi-isometry of solvable minimax groups

[Edits in brackets]
Consider two finitely generated solvable minimax groups $G_i$ ($i = 1,2$) so that $1 \to N_i \to G_i \to Z_i \to 1$ [splits]
with $N_i$ nilpotent, $Z_i$ infinite cyclic and $G_i$ ...

**4**

votes

**0**answers

58 views

### When is the submonoid preserving a subspace finitely generated?

Let $T$ be a topological space with at least one open set whose closure is not open.
Let $G$ be a finitely generated group acting by homeomorphisms on $T$. Let $S\subset T$ be a subspace.
Under what ...

**7**

votes

**2**answers

196 views

### Groups acting on products of hyperbolic spaces

I am interested in groups acting properly and cocompactly by isometries on finite products of Gromov-hyperbolic metric spaces. I am mostly interested in the case where the group itself is not ...

**17**

votes

**1**answer

462 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 ...

**2**

votes

**0**answers

137 views

### Random walk on non-abelian free group

Let $F_2$ be the free non-abelian group with generators $a, b\in F_2$.
Has the "random walk" where we start with the identity and then multiply it by $a$ or $b$ or $a^{-1}$ or $b^{-1}$ ...

**8**

votes

**2**answers

192 views

### Is the automorphism group of free group of rank two relatively hyperbolic?

By Behrstock, Drutu and Mosher [BDM], we know that the (outer) automorphism groups $\mathrm{Aut}(F_n)$ and $\mathrm{Out}(F_n)$ of free group of rank $n$ are not relatively hyperbolic if $n \geq 3$ (...

**2**

votes

**0**answers

91 views

### Calculating the polynomials which are invariant under the action of a simple finite group

Let $G$ be a simple, finite group. In general, $G$ is not abelian.
Let $\rho$ be a representation of this group, where each $\rho(g)$ for $g\in G$ is a unitary, complex, $d$-dimensional matrix, $\rho(...

**1**

vote

**0**answers

94 views

### Rack cohomology as derived functor cohomology

Let $X$ be a rack and $A$ be an $X$-module. By this paper, p. 33, we can associate a cochain complex $C^\bullet(X,A)$ to the pair $(X,A)$. This complex is explicitly defined by a differential $d$. I ...