Questions tagged [gr.group-theory]

Questions about the branch of algebra that deals with groups.

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What is the "permanence relation" really?

I have come across the words "permanence relation" in a 1969 paper by Keith Hannabuss The Dirac equation in de Sitter space. The only other similar google hit for this phrase appears in ...
José Figueroa-O'Farrill's user avatar
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1 answer
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Class number of Burnside groups

Let $B(m,n)$ be the Burnside group on $m$ generators of exponent $n$. Suppose the class number - the number of conjugacy classes - of $B(m,n)$ is finite. Does it imply that $B(m,n)$ is finite?
Meisam Soleimani Malekan's user avatar
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Virtual fibring of $\mathrm{Out}(F_2\times F_2)$

A finitely generated group $G$ is said to virtually fibre if there is a finite index subgroup $H\leq G$ and a non-trivial map $\varphi:H\to\mathbb{Z}$ with $\ker(\varphi)$ finitely generated. I want ...
Marcos's user avatar
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What is known about the upper density of torsion elements in finitely generated groups?

Let $T\subset G$ be the set of all torsion elements in a finitely generated infinite group $G$, and let $B_n\subset G$ be the closed ball of radius $n$ around $1$ w.r.t. to the word metric for some ...
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Spectral sequence construction of Euler class of group extension

Let $A$ be an abelian group equipped with an action of a group $G$ and let $$1 \longrightarrow A \longrightarrow \Gamma \longrightarrow G \longrightarrow 1$$ be an extension of group inducing the ...
Lauren's user avatar
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Aspherical space whose fundamental group is subgroup of the Euclidean isometry group

Let $M$ be a smooth, compact manifold without a boundary, with its universal covering $\tilde{M} = \mathbb{R}^n$. If there exists an injective homomorphism $h: \pi_1(M) \rightarrow O(k) \ltimes \...
Chicken feed's user avatar
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213 views

Quotient of a $F_n$ group which is $F_n$

It is known that quotients of finitely generated groups are finitely generated and that the quotient of a finitely presented group is finitely presented iff the normal subgroup is the normal closure ...
Marcos's user avatar
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Subgroups of the symmetric group and binary relations

Motivation The following came up in my work recently. (NB this is the motivation, not the question I'm asking. You can skip to the actual question below, which is self-contained, but not self-...
Z. A. K.'s user avatar
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Tits indices over $\mathbb{Q}$

Does every Tits index belong to some semisimple algebraic group defined over the field of rational numbers?
Daniel Sebald's user avatar
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Is fundamental group of a finite volume, negatively curved, cusped manifold a non-uniform lattice?

$\DeclareMathOperator\Mob{Mob}$Some background: (1) A Riemannian manifold $M$ is pinched negatively curved if there is a constant $\tau<\kappa<0$ such that all the sectional curvatures are ...
Yanlong Hao's user avatar
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Adjoint identity on finite nilpotent groups

Let $G$ be a finite nilpotent group. Consider the following (adjoint) identity from [BuPa, Theorem 9.6]: $$\left(\prod_{\chi \in \mathrm{Irr}(G)} \frac{\chi}{\chi(1)} \right)^2 =\frac{|Z(G)|}{|G|}\...
Sebastien Palcoux's user avatar
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363 views

Hyperbolic groups and residual finiteness

The existence of a hyperbolic group which is not residually finite is (to my knowledge) an open question. Is there any reason to suspect that all hyperbolic groups are residually finite, perhaps some ...
Mithrandir's user avatar
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The orders of which nonabelian finite simple groups can be written as products of other such orders?

Is it true that the order of a nonabelian finite simple group $G$ can be written as the product of the orders of two or more other nonabelian finite simple groups if and only if $G$ is either an ...
Stefan Kohl's user avatar
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Cohomology of a countable directed union of groups

It's puzzled me for a long time why two arguments in group cohomology look connected but no immediate visible connection is available. First, it is a theorem that if a group $G$ is the union of a ...
Peter Kropholler's user avatar
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Finite groups with number of generators strictly less than number of relations

For the finite cyclic group of order $n$, there is the standard presentation $\langle a \mid a^n\rangle$. Also for $S_n$ (symmetric group), I know a few presentations where the number of relations is ...
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References for completions of finite group tensor categories

Let $G$ be a finite group and $\operatorname{Vec}_G$ be the tensor category of $G$-graded vector spaces (or, if you prefer, $\pi_{\le 2}(BG)$). The completion $\overline{\operatorname{Vec}_G}$ of $\...
Kevin Walker's user avatar
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Groups of non-orientable genus 1 and 2

The non-orientable genus (aka crosscap-number) $\overline{\gamma}(G)$ of a finite group $G$ is the minimum non-orientable genus among all its connected Cayley graphs (and $0$ if $G$ has a planar ...
Kolja Knauer's user avatar
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333 views

Does $\mathit{Suz}$ contain $M_{13}$?

$\newcommand\Suz{\mathit{Suz}}$I recently noticed that the Suzuki group $\Suz$ has as subgroups classes of both $L_3(3)$ and $M_{12}$, both of which are also subgroups of the Mathieu groupoid $M_{13}$....
Daniel Sebald's user avatar
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Terminological question on finite groups

Is there a standard term to denote finite groups $G$ with the property that the projection $\operatorname{Aut} G \to \operatorname{Out} G$ from automorphisms of $G$ to outer automorphisms is split?
Angelo's user avatar
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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 ...
Hussain Rashed's user avatar
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128 views

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\...
jpmacmanus's user avatar
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When is a Schreier coset graph vertex transitive

When is a Schreier Coset graph on a group $G$ with subgroup $H$ and symmetric generating set $S$(without identity) vertex transitive? It is well known that when $H$ is normal, the Schreier coset graph ...
vidyarthi's user avatar
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Which Lie groups are a central extension of an algebraic group?

Suppose $G$ is a connected real Lie group. The quotient $G/Z(G)$ is the image of the adjoint representation, so a linear group. Is it known for which groups this quotient is Lie isomorphic to an ...
Luis's user avatar
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Finitely presentable group with purely infinite full group $C^*$-algebra?

Does there exist an example of a finitely presentable group whose full group $C^*$-algebra is purely infinite, resp. is it known to be impossible?
C-star-W-star's user avatar
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Coxeter's braid group quotients

Coxeter's result is that if the generators of the braid group $B_n$ on $n$ strands fulfill a relation $\forall_i\sigma_i^k=1$, then $1/n+1/k>1/2$ must hold to get a finite quotient of $B_n$. In ...
Hauke Reddmann's user avatar
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0 answers
129 views

Characters of finite groups with Sylow $p$-subgroups of order $p$

Let $G$ be a finite group, $p$ a prime, $P \in Syl_p(G)$ with $|P| = p$ and $C_G(P) = P$. Let $H = N_G(P)$. Assume $H/P \cong C_{p-1}$. Let $\psi \in \text{Irr}(G)$ such that $\psi_H = m \mu$, $m \in ...
Nick's user avatar
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Subgroups of $\mathrm{O}_3$ that are the symmetry groups of compact subsets of $\mathbb{R}^3$

Is there a classification theorem for the subgroups of $\mathrm{O}_3$ that are the symmetry groups of compact subsets of $\mathbb{R}^3$? Apparently, there is an almost complete classification in ...
Arshak Aivazian's user avatar
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216 views

Group cohomology of $\mathbb{Z}$ vs $\mathbb{Z}_p$

Let $M$ be a continuous representation of $\mathbb{Z}_p$ over $\mathbb{F}_p$, likely infinite-dimensional. There is the inflation map of group cohomology $H^*_{\text{cts}}(\mathbb{Z}_p, M) \rightarrow ...
user125639's user avatar
5 votes
0 answers
114 views

Rigid points of $\mathrm{Co}_0$

Let the rigid points of a matrix group refer to subgroups of it that stabilize a nonzero vector and are maximal among such subgroups. How many conjugacy classes of rigid points are there under the ...
Daniel Sebald's user avatar
5 votes
0 answers
191 views

Outer and inner automorphism of $\mathrm{Pin}$ groups

$\DeclareMathOperator\Inn{Inn}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Out{Out}\DeclareMathOperator\Pin{Pin}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}\DeclareMathOperator\PSO{...
Марина Marina S's user avatar
5 votes
0 answers
151 views

Finitely generated nilpotent groups with hyperbolic automorphisms

$\DeclareMathOperator\Out{Out}\DeclareMathOperator\GL{GL}$ Let $G$ be a finitely generated nilpotent group. We call an automorphism of $G$ hyperbolic if the induced automorphism of the free part of ...
Sean Lawton's user avatar
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When is the profinite completion of a Noetherian group ring also Noetherian?

Let $G$ be a group, and let $\mathbb{Z}[G]$ denote its group ring. Its profinite completion is the inverse limit over all ideals of finite index. By Benjamin Steinberg's answer here, this profinite ...
stupid_question_bot's user avatar
5 votes
0 answers
123 views

Applications of $FP_\infty$ groups preserving direct systems

In [1], the author proves that given a group $G$ and a directed system $(M_\lambda)_\lambda$ of $G$-modules, the induced maps $$\varinjlim H^k(G,M_\lambda) \to H^k(G,\varinjlim M_\lambda)$$ are ...
Mark Backhaus's user avatar
5 votes
0 answers
163 views

Finite simple groups of automorphisms of finite simple Lie algebras

I begin by briefly recalling some basic facts in order to pose my question in context. According to the classification, the finite simple groups are cyclic of prime order, are alternating on $n \geq 5$...
user203598's user avatar
5 votes
0 answers
112 views

Computability of the "free envelope rank" of an endomorphism of a free group

Let $F$ be a free group freely generated by the finite set $S$ and $\sigma\colon F\to F$ be a group morphism. We define the free envelope rank of $\sigma$, written $r(\sigma)$, as the smallest $k$ for ...
user158448's user avatar
5 votes
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202 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 ...
alvarezpaiva's user avatar
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A lemma concerning conjugations and normal subgroups (related to a theorem of Frobenius)

In the paper "On A Theorem of Frobenius" written in 1969, Prof. Richard Brauer, for the first time, presented a character-free proof to the Frobenius theorem (i.e. counting the number of ...
Topoiii's user avatar
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0 answers
192 views

Tools for computing from group presentations

What are some tools -- either theoretical/by hand or algorithmic/by computer -- that are useful for doing computations in finitely presented groups? In my particular case, I'm working with a finitely ...
Ethan Dlugie's user avatar
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Groups whose exponential growth is never pinched

This question is motivated by a partial answer to another question on MO. Given an infinite finitely generated group $G$ and a finite generating set $S$, let $b_n^S$ be the cardinality of the ball of ...
ARG's user avatar
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Which tensor power of a given representation contains the trivial one?

If $R$ is an irreducible representation of a simple Lie-groups $G$ I assume there is always a lowest integer $n$ such that the tensor product representation $R \otimes R \otimes \ldots \otimes R$ (n ...
Fetchinson0234's user avatar
5 votes
0 answers
184 views

About the elements of bounded order in finite groups

Let $G$ be a finite group and $n$ be a positive integer. Define $$X_n(G):=\{x\in G: x^n=1\}.$$ Define $$J_n=\left\{\frac{|X_n(G)|}{|G|}: G\,\text{is a finite group}\right\}.$$ Question. Does there ...
Meisam Soleimani Malekan's user avatar
5 votes
0 answers
123 views

Conjugacy classes of plane k-jet group

Define $G(n, k)$ as a subgroup of $\rm{Aut}(\Bbb C[[x_1, \dots, x_n]]/\mathfrak m^{k+1})$ with identity linear part (so, group of $k$-jets of selfmaps of $\Bbb C^n$). I'm interested in the map from $G(...
Denis T's user avatar
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5 votes
0 answers
191 views

Monster group as automorphism group of a distributive lattice

It is known that every finite group is the automorphism group of a finite distributive lattice. Question: What is the minimal order of a distributive lattice $L$ such that the automorphism group of $...
Mare's user avatar
  • 25.8k
5 votes
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280 views

Matrix groups with two generators

Given two matrices $A,B\in{\rm{SL}}_2(\Bbb{R})$, is there any criterion guaranteeing that the subgroup they generate is discrete? What if one puts restrictions on $A,B$ e.g. they are both elliptic? ...
KhashF's user avatar
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About a diophantine equation from group theory

Is there any set of odd primes $\{p_1, p_2,..., p_k\}$ and natural numbers $a_1,..., a_k$ such that the following equation satisfied: $${p_1^{2a_1+1}+1 \over p_1+1}\times ....\times {p_k^{2a_k+1}+1 \...
BHZ's user avatar
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5 votes
0 answers
343 views

Applications of Tits' alternative in algebraic number theory

I have recently studying Tits' alternative. The theorem statement goes like the following: Tits' alternative: Let $G$ be any finitely generated linear group over a field. Then one of the following is ...
ShBh's user avatar
  • 271
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171 views

Are the symmetric groups integrable as Hopf algebras?

Let $G$ be a group. For $g,h \in G$, let $[g,h]=g^{-1}h^{-1}gh$ be a commutator. The normal subgroup $G' = \langle [g,h] \ | \ g,h \in G \rangle$ is called the commutator subgroup or derived subgroup. ...
Sebastien Palcoux's user avatar
5 votes
0 answers
140 views

Reference request: Name or use of this group of diffeomorphisms of the disc

Let $k \in \{0,\infty\}$, $G\subseteq \operatorname{Diff}^k(D^n)$ be the set of diffeomorphisms $\phi:D^n\to D^n$ of the closed $n$-disc $D^n$ (with its boundary) satisfying the following: $ \phi(S_r^...
ABIM's user avatar
  • 5,019
5 votes
0 answers
187 views

Number of elements in $\mathrm{GL}(n,p)$ with maximal order

I learned reading this question that $\mathrm{GL}(n,p)$ elements have at most a multiplicative order of $p^n -1$. I would like to know how many matrices have an order of exactly $p^n -1$. Do they ...
Cyrius Nugier's user avatar
5 votes
0 answers
257 views

Nimber $2^{2^k} - 1$ is a multiplicative generator of $[2^{2^k}]$?

Let $t = 2^{2^k}$, and consider the field $[t]$ of nimbers below $t$. For $k \leq 6$ one can check that $t - 1$ (in the usual arithmetic sense) is a multiplicative generator of $[t] \backslash \{0\}$. ...
Mikhail Tikhomirov's user avatar

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