Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
2,078
questions with no upvoted or accepted answers
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Character tables of finite groups and isomorphism
I'd like to ask the following question:
Let $G$ and $H$ be finite groups.
Is there a useful criterion involving the ordinary character table which assures that $G$ and $H$ are isomorphic as groups?...
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Lattices of minimal covolume in $\operatorname{SL}_2(\mathbb{R}) \times \operatorname{SL}_2(\mathbb{R})$
What are the (uniform/non-uniform) irreducible lattices of minimal (or even small) covolume in $\operatorname{SL}_2(\mathbb{R}) \times \operatorname{SL}_2(\mathbb{R})$?
Context: Such a lattice will ...
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A finitely presented infinite group where every non-trivial element is conjugate to a power of a fixed element
Does there exist a non-abelian finitely presented infinite group $G,$ with a non-trivial element $y\in G$ such that every non-trivial element $x\in G$ is conjugate to some power of $y$? More ...
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Examples of non-uniform lattices in products of trees
Consider a product of two locally-finite, infinite, unimodular trees $X=T_1\times T_2$. Assume that both ${\rm Aut}(T_1)$ and ${\rm Aut}(T_2)$ are not discrete.
So as a vague general question, what ...
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How much choice is required for a countably-infinite index subgroup of the real additive group?
The existence of such subgroups implies the existence of a non-measurable set; simply intersect each of the cosets with $[0,1]$. The results will all have equal outer measure, but their union will be ...
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203
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Universal group on $\kappa$ elements
It is well known that for every positive cardinal $\kappa$, every group of cardinality $\kappa$ can be embedded into $\text{Sym}(\kappa)$, the group of bijections on $\kappa$ with composition as group ...
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291
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Writing canonically a transitive action as quotient of a simply transitive action
Consider a finite group $G$ acting transitively on a finite set $Y$. Is it possible to find a finite set $P_Y$ and a finite group $\hat G$ acting on $P_Y$ such that the following hold?
The action of $...
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Is there an equivariant simplicial deformation retract of Teichmüller space?
Let $S_g$ be a surface of genus $g \ge 2$. By analogy with Teichmüller space for $S_g$, Culler and Vogtmann studied Outer Space $CV_n$, with points projective classes of marked metric graphs with ...
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188
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Description of quasimorphisms of the free group
Let $F$ be a free group of finite rank with a fixed basis and corresponding word metric. Let $Q = Q^0_h(F, \mathbb{R})$ be the space of real homogenous quasimorphisms that vanish on the basis of $F$. ...
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Explicit description of the smallest class of groups, that contains all finite simple groups and is closed under semidirect products
Suppose $\Pi$ is the smallest class of groups satisfying the following conditions:
All finite simple groups lie in $\Pi$
If $G \cong H \rtimes K$ and both $H$ and $K$ are in $\Pi$, then $G$ is also ...
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196
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Are finite groups of exponent $d$ rare for $d \neq 4$?
Is there a way to prove, that $\lim_{n \to \infty} \frac{\text{the number of all groups of exponent }d \text{ and order less than }n}{\text{the number of all groups of order less than } n} = 0$ for $d ...
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Finitely generated nilpotent groups as cusp groups
I recently learned about the following question, asked by I. Kapovich :
Is there an example of a group $G$ which is hyperbolic relative to some parabolic subgroups that are nilpotent of class $\geq 3$...
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Continuity of the Green function with respect to the measure
Let $G$ be a finitely generated group and let $\mu$ be a finite measure on $G$. Define the Green function as
$$G(\mu)=\sum_{n\geq 0}\mu^{*n}(e),$$
where $\mu^{*n}$ is the $n$th convolution power of $\...
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357
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Generalization of a theorem of Frobenius
If $G$ is a finite group, $H\leq G$ a subgroup, and $n$ divides $|G|$, then $n$ divides $[G:N_{G}(H)]$ times the number of elements $g\in G$ for which $|\langle H,g\rangle|$ is a divisor of $n$.
...
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Are there torsion-free restricted simple Lie algebras?
It is known that a torsion-free group can be simple (see e.g. Rataggi's paper https://www.degruyter.com/abstract/j/jgth.2007.10.issue-3/jgt.2007.028/jgt.2007.028.xml). I would like to know if the ...
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Polish groups with no small subgroups
Definitions.
A Polish group is a topological group $G$ that is homeomorphic to a separable complete metric space.
A group $G$ has no small subgroups if there exists a neighborhood $U$ of the identity ...
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Group with Character Degrees {1,pq,pr,qr}, where p,q and r are distinct primes
I am currently trying to bound the derived length of certain solvable groups assuming that they have only two irreducible monomial complex character degrees. Using induction, it often suffices to ...
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Uniform versus non-uniform group stability
Group stability considers the question of whether "almost"-homomorphisms are "close to" true homomorphisms. Here, "almost" and "close to" are made rigorous using a group metric.
More precisely, ...
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Any f.p. faithful simple module over a primitive group ring?
Recall that a ring $R$ is primitive if it has a faithful simple left module. Let $G$ be a countable discrete group and $R=\mathbb{k}G$, where $\mathbb{k}$ denotes some field or $\mathbb{Z}$.
There ...
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Do the ternary braid groups arise in algebraic topology?
Let $TB_{n}$ be the group defined by the presentation with generators $t_{1},...,t_{n-2}$ and relations $t_{i}t_{i+1}t_{i+2}t_{i}=t_{i+2}t_{i}t_{i+1}t_{i+2}$
and $t_{i}t_{j}=t_{j}t_{i}$ whenever $|i-j|...
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Non-algebraic quasi-isometric embeddings
What are examples of finitely generated groups $\Gamma$ and $\Lambda$ such that the metric space $\Lambda$ embeds into $\Gamma$ quasi-isometrically but such that $\Lambda$ is very much not a subgroup ...
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215
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Automorphism groups of cocompact Fuchsian groups as mapping class groups
Let $\Gamma$ be a cocompact Fuchsian group. So it has presentation $$\langle x_1,y_1, \dots, x_g,y_g,z_1, \ldots, z_r \mid [x_1,y_1] \cdots [x_g,y_g]z_1 \cdots z_r=1, \ z_i^{m_i}=1 \rangle$$
for some $...
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Uniqueness of the boundary of a hierarchically hyperbolic group
Hierarchically hyperbolic groups and spaces (HHG and HHS for short) were defined by Behrstock, Hagen and Sisto (see here and here). Examples include mapping class groups, Right angled Artin groups, ...
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Are these element in a group algebra of a torsion-free group zero divisors?
Let $G$ be an arbitrary torsion-free group. For $x,y\in G$, which of these elements can be decided immediately not to be zero divisors in $\mathbb ZG$ (or in $\mathbb CG$)?
$$1+x+y,\quad 4+x+x^{-1}+y+...
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The preimage of bounded real intervals under homomorphisms on hyperbolic groups
Let $G$ be a hyperbolic group with a fixed (finite, symmetric) generating set and suppose that $\varphi : G \to \mathbb{R}$ is a group homomorphism. Write $W_n = \{ g \in G: |g|=n\}$, where $|g|$ ...
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A Spatial-Orientation Counting Problem
Suppose I have 36 black blocks of dimensions 1x2x3. I can stack them 2 across, 3 deep and 6 high to make a nice looking cube of dimensions 6x6x6. I then proceed to paint the surface of this cube red. ...
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Projecting GxG onto subspace with tied irreducible representations
Suppose I have a finite group $G$. With this group, I can associate an ortho-normal Hilbert space spanned by elements of the group $$\mathcal{H} = \{|g\rangle: g \in G \}$$.
I could alternatively ...
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303
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Central extension of Tarski monsters
Suppose $G$ is a perfect group ($G=G'$) with the following properties. $G/Z(G)$ is a Tarski $p$-group or another simple finitely generated infinite group in which all proper subgroups are abelian, and ...
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134
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Filling points to a simplex in models for EG
I have a question which is related to higher Dehn functions of groups.
I also have a group $G$ with a finite $K(G,1)$. Let us denote by $EG$ the universal cover of this complex. We choose a path-...
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113
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$m$-thick sets with small $n$-fold sumsets in finite cyclic groups
Problem. Is it true that for every positive integers $n,m$ there exists a subset $A_{n,m}$ of a finite cyclic group $G$ having the following two properties:
$(\Sigma_n)$ the $n$-fold sum $A_{n,m}^{+...
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An example of a residually-finite group which does not have an infinite residually-$p$ subgroup
Does there exist a residually-finite group $G$, preferably finitely generated, such that $\widehat{G}$, the profinite completion of $G$, contains an infinite pro-$p$ subgroup, but $G$ does not contain ...
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Maximal subgroups of infinite index and profinite completion
Preliminary remark: I'm mainly interested in an answer (or link to ressources) in the specific context of the first Grigorchuk group, but I believe that it may be of some interest to state the ...
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Divisible orientation preserving diffeomorphism which is time-$1$ map of no smooth flow
Is there an orientation preserving smooth diffeomorphism $f$ on a compact manifold $M$ such that
for every $n\in \mathbb{N}$, there is a smooth diffeomorphism $g:M \to M$, as $n$th root of ...
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subgroups of $\mathrm{Sp}_{2g}(\mathbb{Z}_2)$ whose mod-2 image is the symmetric group
Let $G \subseteq \mathrm{Sp}_{2g}(\mathbb{Z}_2)$ be a closed subgroup of the symplectic group over the $2$-adic integers whose image under the mod-$2$ homomorphism $\pi : \mathrm{Sp}_{2g}(\mathbb{Z}_2)...
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Is there a composite-order generalization of the homomorphism on Rep(Z/p) giving total dimension of Tate cohomology?
Let $p$ be a prime, let $\mathbb{Z}_p$ be the ring of $p$-adic integers, and let $G$ be a cyclic group of order $p$. It is rather well-known that finite rank $\mathbb{Z}_p$-free representations of $G$...
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Intermediate subgroups between $H(\mathbb{Z})$ and $H(\mathbb{Z}[\sqrt{2}])$, for anisotropic form of $SL_2$
Let $H$ be an anisotropic $\mathbb{Q}$-form of $SL_2$, with $H(\mathbb{R}) = SL_2(\mathbb{R})$.
Are there any results or conjecture regarding the existence, or non-existence of intermediate subgroups ...
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normal form for some finite groups, extending the small groups library
I am in need of a normal (that is, canonical) form for (some) finite groups, computable with - for example - gap or sage or any other freely available package. The goal is to make finite groups ...
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Are the centralizers of involutions in finite simple groups known?
By the famous result of Brauer and Fowler, there exist finitely many simple groups with given involution centralizer. There are many results which determine all finite simple groups with given ...
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Coarsely Lipschitz retractions onto cyclic subgroups
A good way to show that a subspace is undistorted is to give a coarse Lipschitz retraction of the whole space onto that subspace. This question is about a failure of the converse.
Let $G$ be a ...
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Golod-Shafarevich groups and L_2- Betti numbers
Is it something known about $L^2$-Betti numbers for Golod-Shafarevich groups?
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A question about retracts of a group
A group $H$ is called a retract of a group $G$ if there exist homomorphisms $f:H\longrightarrow G$ and $g:G\longrightarrow H$ such that $g\circ f=id_H$. By a trivial retract of $G$, I just mean the ...
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167
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In the literature on infinite graphs, are there results on "periodizable" graphs?
Let $G=(V,E)$ be a connected countably infinite $k$-regular simple graph (no loops or multiple edges). For $A$ a finite subset of $V$, let me denote by $G_A=(A,E_A)$ the induced subgraph with vertex ...
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116
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Can we describe the nonabelian exterior square of a finite 2-generated metabelian group?
Let $G$ be a finite 2-generated metabelian group, and let $S$ be a schur covering group, so that we have an exact sequence
$$1\rightarrow M(G)\rightarrow S\rightarrow G\rightarrow 1$$
where $M(G)$ is ...
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modularity of Theta functions attached to Hecke characters
Let $K/\mathbb{Q}$ be a quadratic imaginary field, and let $\chi$ be a Hecke character on $K$. Using Poisson summation, one can show that the theta function
$$
\theta(z):=\sum_{I\subseteq \mathcal{O}...
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Isospectrality, Gassmann-Sunada triples, and tensor products
It is well known that Gassmann-Sunada group triples can be used to construct isospectral manifolds, arithmetically equivalent number fields, etc. (Recall that a Gassmann-Sunada triple $(U,V,W)$ ...
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432
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Subgroups and quotients of an abelian pro-finite group
It is well known that every subgroup $H$ of a finite abelian group $G$ is isomorphic to a quotient of $G$.
I'm wondering whether there is a counterpart for profinite groups.
For example is it true ...
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163
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The group of automorphisms a pair $(G,X)$ where $X$ is a spherical homogeneous space of $G$
I wish to "compute" ${{\rm Aut}}(G,X)$ for a spherical homogeneous space $X$ of $G$ in terms of the spherical datum of $X$.
First let $G$ be any algebraic group over $\mathbb C$, and let $X$ ...
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519
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Atlas of finite groups, Character table of automorphism group of sporadic group
I am consulting ATLAS of finite group for character table of Automorphism Group of sporadic group.
I am reading from Inverse Galois Theory by G. Malle
Let me start with $G=M_{12}$
This(image ...
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112
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Do the "Nielsen" IA-automorphisms of a profinite free group $\widehat{F}$ of rank 2 form a normal subgroup of $\mathrm{Aut}(\widehat{F})$?
Let $F$ be the discrete free group of rank 2, and let $\widehat{F}$ be its profinite completion, equipped with an embedding
$$i : F\hookrightarrow\widehat{F}$$
By a result of Asada, this embedding ...
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238
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Counting the number of orbits finite groups of "diagonal type"
Let $n$, $k$, $r_1, \dots, r_k$ be positive integers.
For each $i \in [k]:=\{1,\dots,k\}$, suppose we are given $n$ permutations of the the set $[r_i]$, that is $f_1^{(i)}, \dots, f_n^{(i)}$ in $\...