Questions tagged [gr.group-theory]

Questions about the branch of abstract algebra that deals with groups.

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50 views

Number of commuting pairs in p-group

Let $p$ be a prime number and $G=GL_n ( \mathbb{Z} / p \mathbb{Z} )$. Consider the set $U$ of upper-triangular matrices of $G$ having entries of $1$ on the diagonal. The set $U$ is a Sylow $p$-...
1
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0answers
22 views

Does Levi operator always map one-word varieties to one-word varieties?

Suppose $\mathfrak{U}$ is a group variety. Now let’s define $L(\mathfrak{U})$ as the class of all such groups $G$, such that $\forall g \in G$ $\langle \langle g \rangle \rangle \in \mathfrak{U}$ (...
7
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0answers
87 views

Which irreducible representations of the symmetric group are eigenspaces of class sums?

In the setting of complex representations of finite groups, a class sum $1_C=\sum_{g\in C} g$ acts on an irreducible representation $V$ as $\lambda(C,V)\operatorname{Id}$, where $\lambda(C,V)=|C|\...
1
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0answers
105 views

For a finite subgroup of an infinite group $G$, does its normal closure have infinite index in $G$?

For a finite subgroup of an infinite group $G$, does its normal closure have infinite index in $G$? Or if not, is it true when we replace $G$ by some subgroup? That is: Let $H$ be a finite ...
4
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2answers
161 views

Mackey theory in the setting of locally profinite groups

$\DeclareMathOperator\Hom{Hom}$Let $R$ be a commutative ring (not necessarily unital). Let $G$ be a finite group, and let $H_1, H_2$ be subgroups of $G$. Recall the following standard result [1, Thm. ...
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0answers
93 views

Is there exists a lattice isomorphism?

Let $\text{P}$ be the set of partitions of {1,2,...,n} and $\text{Y}$ the set of Young subsets of permutation group S(n)(the coxeter group of type An). As is well-known, the set $\text{Y}$ is a ...
4
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0answers
119 views

Steinberg relations for elementary subgroup of a Chevalley group over an arbitrary ring

Given a semisimple Lie algebra $\frak{g}$ of type $\Phi$ with a Lie algebra representation $\rho:\frak{g}\to \frak{gl}(v)$ and an arbitrary commutative ring one can associate the following gadgets: ...
4
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1answer
137 views

Uniform Roe algebra of virtually abelian group is type I C*-algebra?

Let $G$ be an arbitrary (discrete) group. It acts by left translation on $\ell^\infty(G)$. The uniform Roe algebra of $G$ is defined as the crossed product $\ell^\infty (G) \rtimes_{\mathrm{red}}G$. ...
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1answer
109 views

$G$- space is locally compact [on hold]

Suppose $X$ is a topological space ,$G$ Is a locally compact group.If the quotient space $G\backslash X$ is compact,can we deduce that $X$ is locally compact?
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26 views

Do all maximal verbal series have the same length?

Suppose, $G$ is a finite group. Let’s call a series of subgroups of $G$ $\{H_k\}_{k=1}^n$ a verbal series of length $n$, iff $H_1 = E$, $H_n = G$ and $\forall k < n$ $H_k$ is a verbal subgroup of $...
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145 views
+50

Subset of reals associated to pairs of matrices in $\mathrm{SL}(2,\mathbb{R})$

Let $\Gamma$ be a subgroup of $\mathrm{SL}(2,\mathbb{R})$. I would like to ask if there is any research on the following set: $$\Gamma*\Gamma:=\bigg\{\dfrac{(a+b)(a'+b')}{(c+d)(c'+d')}\bigg|\begin{...
5
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0answers
179 views

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 ...
7
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1answer
288 views

Gromov hyperbolic groups which are solvable are elementary

I have read on wikipedia that a Gromov hyperbolic group which is solvable is elementary (i.e. virtually cyclic). Where can I find a proof of this fact? There is a proof of a similar fact in Bridson-...
4
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0answers
68 views

Finite groups of cyclicality index $3$

Suppose $G$ is a group. Let’s define the cyclicality index of $G$ using the following recurrent relation: $$CI(G) = \begin{cases} 1 & \quad G \text{ is cyclic} \\ \max_{H < G} CI(H) + 1 & \...
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0answers
58 views
+50

Efficiently computable graph conductance measures

Treating electrical networks from a graph theory point of view, do there exist measures that characterize the overall electrical conductance of the graph whilst being efficiently computable? There ...
12
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2answers
322 views

Does asymmetric fraction of finite groups tend to $0$?

Let’s define asymmetric fraction of a finite group $G$ as the number $$\mathrm{af}(G) = \frac{|\{(g, a) \in G \times \mathrm{Aut}(G)\mid a(g) = g\}|}{|G|\cdot|\mathrm{Aut}(G)|}.$$ Equivalently it can ...
4
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0answers
63 views

Amalgamated subgroup of an HNN extension finitely generated

Baumslag proved that if $G= A \ast_{C} B$ is an amalgamated free product where $A$ and $B$ are finitely presented, $G$ is finitely presented if and only if $C$ is finitely generated. Similarly, by ...
6
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1answer
258 views

Wreath product $S_k\wr S_n$ inside $S_{kn}$

I want to understand wreath products a little better. Currently, my intuition about them is as follows. Take $nk$ disks, pile them in order forming $n$ piles of $k$ disks (one pile contains disks {1,...
5
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0answers
131 views

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 $...
-1
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1answer
78 views

Union of an ascending chain of subgroups in group $G$ isomorphic to subgroup $S_0\subseteq G$ [closed]

Let $G$ be an infinite group, let $S_0\subseteq G$ be a subgroup and suppose that ${\frak C}$ is a collection of subgroups of $G$ such that $C \cong S_0$ for all $C\in {\frak C}$, and for all $C, C'\...
4
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0answers
161 views

Algebraic varieties associated to finite groups

Have the following equations been studied in the literature? Let $G$ be a finite group. Then I am looking for functions $f : G \rightarrow \mathbb{C}~ \backslash \left\lbrace 0 \right\rbrace $ such ...
3
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1answer
55 views

Is $S_\omega/F_\omega$ embeddable to $S_\omega$?

Let $S_\omega$ be the group of bijections $f:\omega\to\omega$, and let $F_\omega = \{\pi\in S_\omega: \exists N\in \omega(\pi(k) = k \text{ for all } k\geq N)\}$. It is easy to see that $F_\omega$ is ...
8
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0answers
287 views

Can the set of endomorphisms of $(\mathbb R,+)$ have cardinality strictly between $\frak c$ and $\frak{c^c}$?

Let $\frak c$ be the cardinality of the reals. I know that in ZF the set of endomorphisms of $(\mathbb R,+)$ can have at least two different cardinalites: If we allow the axiom of choice, you can ...
20
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3answers
2k views

In what sense is SL(2,q) “very far from abelian”?

I am far from an expert in this area. I'd be grateful if someone could explain in what sense $\mathop{SL}(2,q)$ is "very far from abelian," to quote Emanuele Viola? Why does Theorem 1 (below) justify ...
0
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1answer
116 views

Number of cycles under a certain action on Z/nZ [closed]

Computer scientist here looking at a question that came about from in-place matrix transposition, but rusty on my abstract algebra and number theory... Suppose we have the multiplicative group $\...
2
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0answers
89 views

On 2 crossed modules

Let $(G,H,\alpha,\tau)$ and $(H,J,\alpha',\tau')$ be two crossed modules of groups. It is given that $Kernel(\tau) \cap Image(\tau')$ =$e$. For every $h\in H$ does there always exist a $j_h \in J$ ...
5
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0answers
152 views

How bad can the orbit space (assume it‘s finite) be?

Fix a connected algebraic agroup $G$ over a local field $F$ (e.g real numbers). Consider all varieties $X$ over $F$ with $G$ action such that $O_X=X(F)/G(F)$ is finite. Using anayltic topology $X(F)$ ...
2
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0answers
138 views

Finite index subgroup of HNN extension

Let $GX$ be a tree group (a right-angled Artin group such that the graph is a tree), such that not all the subgroups of $GX$ are necessarily RAAGs, so the length of the tree is greater or equal than $...
4
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0answers
86 views

When are extensions of algebraically good groups algebraically good?

Let $G$ be a discrete group. The pro-algebraic completion of $G$ is a pro-algebraic group $G^{\mathrm{alg}}$ together with a morphism $s:G\to G^{\mathrm{alg}}$ which is initial among all morphisms ...
2
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0answers
69 views

Spacetime symmetries

We know some nice space-time have a lot of symmetries. It is said that Minkowski spacetime has $$ISO(d-1,1)/SO(d-1,1),$$ de Sitter spacetime has $$SO(d,1)/SO(d-1,1)$$ and anti-de Sitter spacetime ...
5
votes
1answer
145 views

Anti-holomorphic involutions of a complex linear algebraic group

Let $G$ be a connected linear algebraic group over the field of complex number ${\Bbb C}$. Let $G({\Bbb C})$ denote the complex Lie group of ${\Bbb C}$-points of $G$. Let $\sigma$ be an anti-...
3
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1answer
113 views

Commensurator of a subgroup of matrices

Let $k$ be a totally real number field and let $\mathcal{O}_k$ denote its ring of integers. If $H$ is a subgroup of $\text{GL}(n, \mathbb{R})$ let denote with $H(k)$ and $H(\mathcal{O}_k)$ the ...
5
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0answers
96 views

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 ...
4
votes
0answers
91 views

Generating the monoid of injections of the free group into itself

Let $F$ be the free group of rank $2$ (or any finite rank if this does not matter). The set of injections $F\to F$ forms a monoid $M$ by compositions. Is there a simple looking set of generators of $M$...
1
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1answer
117 views

Diagonal automorphisms for twisted Chevalley groups

Let $G$ be a Chevalley group over a field $k$ of characteristic $0$. We know that a diagonal automorphism $\phi_h$ of $G$ is of the form $g\mapsto hgh^{-1}$, where $h\in \hat H$ and $\hat H$ ...
1
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0answers
73 views

On normalized 2-cocycle

Let $G$ be a group acts trivially on an abelian group $A$. Let $\varepsilon $ be a normalized 2-cocycle in $ Z^{2}(G,A)$. Assume that $G=H_{1} \times H_{2}$ and let $\varepsilon_{1}=res_{H_{1}\times ...
7
votes
2answers
132 views

Existence of n-axial elements in groups with at least 2 ends

Let $G$ be a finitely generated group. Fix some symmetric finite generating set $S$ for $G$, and write $\Gamma$ for the Cayley graph of $G$ with respect to $S$. Given finite subsets $X,S,Y$ of $G$, ...
5
votes
2answers
295 views

Is random walk drift rational?

(Question mildly edited for clarity by request of Matt F.) If $G$ is a finitely presented group, let $|\cdot|$ denote the word metric with respect to a finite set of generators. Suppose $\nu$ is a ...
6
votes
0answers
164 views

Diameter of finite rational matrix groups

Suppose $G$ is a finite subgroup of $\mathrm{GL}(n,\mathbb{Q})$. For a set $\mathcal{M} \subseteq G$ that generates $G$, define the $\mathcal{M}$-diameter $\mathit{diam}(G, \mathcal{M})$ of $G$ to be ...
3
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1answer
142 views

Cohomology of linear algebraic groups

Let $R$ be a commutative ring. Let $G\subset \mathrm{GL}_m$ be a linear algebraic subgroup. Has the group cohomology $H^i(G(R),R^m)$ been studied in the literature? For example, do we know (1) $H^...
2
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1answer
89 views

Primitive action of wreath product

I say in advance that I am really new to Group Theory, so if my question is trivial I apologize in advance. Let $A$ and $H$ be groups and $\Omega$ be a $H$-set. In this set-up, we can define the ...
1
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0answers
83 views

Change variable in integration with symmetry

Not sure if I can ask such fundamental problem here. Let $G$ be a finite group, $\sigma \in G$. Consider linear group actions fo $G$ on $\mathbb{R}^n$. $\sigma(K)=\{x\in \mathbb{R}^n: \sigma^{-1}(...
5
votes
0answers
109 views

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 $F$. ...
5
votes
1answer
186 views

Relations between boundaries of groups acting on hyperbolic spaces with WPD elements

Let $(X,d)$ be Gromov-hyperbolic space and let $\Gamma$ be a finitely generated group acting on $\Gamma$ by isometries. Recall the following two definitions. Say that the action is acylindrical if ...
4
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0answers
66 views

How are reflection groups related to general point groups?

I always tried to understand how the finite reflection groups of $\Bbb R^d$ (of some fixed dimension $d$) relate to the point groups of the same space $\smash{\Bbb R^d}$ (finite subgroup of the ...
1
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0answers
50 views

Number of conjugacy classes of unit groups of modular group algebras

Let $n$ be a natural number, $p$ a prime number, $G$ a finite $p$-group and $K$ a finite field with $p^n$ elements. We focus on the group $1+J(KG)$, where $J(KG)$ is the Jacobson radical of $KG$, ...
1
vote
1answer
125 views

Is it possible to extend this homomorphism?

Let $G$ be a torsion free group and $\alpha$ be a non-zero element in its complex group algebra. Assume that $\mathfrak A$ is the Banach sub-algebra of $\ell^1(G)$ generated by $\alpha$. Is it ...
4
votes
0answers
84 views

Do positive-density subgroups intersect nontrivially?

Let $G$ be an infinite finitely generated group and $S$ a generating set. Define density with respect to the sequence of balls $S^n$. If $H_1, H_2 \leq G$ have positive density, must $H_1 \cap H_2$ ...
4
votes
0answers
188 views

Number of labeled Abelian groups of order n [migrated]

I calculated the number of labeled Abelian groups of order $N$ (i.e., the number of distinct, abelian group laws on a set of $N$ elements). This sequence is given by OEIS A034382, but my solution ...
13
votes
1answer
381 views

Dimensional gap of group representations

The problem is inspired by eigenvalue bounds of random Cayley graphs on $SL_2(q)$. Definition. An infinite series of finite groups $S$ is α-rich if the dimension of the smallest nontrivial ...