Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
2,077
questions with no upvoted or accepted answers
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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 ...
6
votes
1
answer
230
views
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?
5
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189
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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 ...
5
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202
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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 ...
5
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143
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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 ...
5
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0
answers
236
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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 \...
5
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213
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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 ...
5
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195
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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-...
5
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0
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117
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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?
5
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0
answers
139
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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 ...
5
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345
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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|}\...
5
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363
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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 ...
5
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0
answers
134
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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 ...
5
votes
0
answers
152
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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 ...
5
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185
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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 ...
5
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77
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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 $\...
5
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193
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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 ...
5
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333
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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}$....
5
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181
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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?
5
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0
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125
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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 ...
5
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128
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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\...
5
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0
answers
185
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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 ...
5
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198
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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 ...
5
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0
answers
110
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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?
5
votes
0
answers
225
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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 ...
5
votes
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answers
129
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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 ...
5
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answers
163
views
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 ...
5
votes
0
answers
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 ...
5
votes
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answers
114
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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 ...
5
votes
0
answers
191
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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{...
5
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answers
151
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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 ...
5
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answers
218
views
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 ...
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123
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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 ...
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163
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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 $n \geq 5$...
5
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answers
112
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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 ...
5
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0
answers
202
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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 ...
5
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0
answers
405
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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 ...
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192
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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 ...
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0
answers
143
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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 ...
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0
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253
views
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 ...
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answers
184
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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 ...
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answers
123
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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(...
5
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0
answers
191
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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 $...
5
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0
answers
280
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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? ...
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337
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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 \...
5
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answers
343
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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 ...
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171
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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. ...
5
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0
answers
140
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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^...
5
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0
answers
187
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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 ...
5
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0
answers
257
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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 $[t] \backslash \{0\}$. ...