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Questions tagged [gr.group-theory]

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

7
votes
1answer
104 views

Topological amenability vs amenability of an action

Let $G$ be a discrete group and let $X$ be a compact, Hausdorff space. Assume that $G$ acts on $X$ by homeomorphisms. Consider the following two definitions: [$C^*$-algebras and finite dimensional ...
6
votes
2answers
98 views

Finite groups with few conjugacy classes of maximal subgroups

Let $c$ be a positive integer, $G$ a finite group with at most $c$ conjugacy classes of maximal subgroup. What can we say about $G$? Same question, but this time $G$ is a finite group with at most $c$...
7
votes
1answer
314 views

Simple groups of the same order

I heard that there are no 3 nonisomorphic simple groups of the same order. Question: Is there an elementary proof of this? In case this is not the case, here a modified question: Question: Is ...
0
votes
0answers
52 views

Coxeter group action on the product of root systems

Let W be a finite Coxeter group and $\Phi^+$ the set of its positive roots. The Coxeter group acts on $\Phi^+$ by $(w, \alpha) \mapsto w \cdot \alpha$ if $w \cdot \alpha \in \Phi^+$ and $(w, \alpha) \...
4
votes
2answers
174 views

Is the *unreduced* Burau representation unitary?

In a 1984 paper, Squier wrote down an explicit Hermitian form $J$ relative to which the reduced Burau representation is unitary. The form is valued in the ring $\mathbb{Z}[s^{\pm 1}]$, where $s^2 = t$ ...
3
votes
0answers
197 views

A new combinatorial problem for finite groups

In a recent preprint arXiv:1811.10503, I proved that if $a_1,\ldots,a_n$ are distinct elements of a torsion-free additive abelian group $G$, then there is a permutation $\pi\in S_n$ such that all ...
2
votes
0answers
40 views

Properties of extendable irreducible characters to a normal Sylow subgroup

Let $G$ be a finite group with a nilpotent commutator subgroup, and denote by $mcd(G)$ the set of degrees of the irreducible monomial characters. Suppose $|mcd(g)|=2$. Furthermore, we call a monomial ...
1
vote
0answers
64 views

On $n$th class-preserving automorphism of finite $p$-group

Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. An automorphism $\alpha$ of $G$ is called an $n$th class-preserving if for each $x\in G$, there exists an element $g_x\in \gamma_n(G)$ ...
5
votes
1answer
178 views

Linear representation of the free metabelian / 2-step nilpotent profinite groups on 2 generators

Let G be the free profinite group on 2 generators, $A=G/[G,[G,G]],B=G/[[G,G],[G,G]]$, then what is the structure of the groups $A$ and $B$? I heard that $A$ is isomorphic to the group of such ($3\...
6
votes
0answers
164 views

A challenging problem on disjoint cosets

Motivated by my negative solutions (see this paper published in Chinese Ann. Math. 13A(1992)) to two open problems on disjoint residue calsses posed by A. P. Huhn and L. Megyesi [Discrete Math. 41(...
0
votes
1answer
84 views

class structure constants relation

Let $C_{j,k}^l$ ,usually called class structure constants, eg Jansen and Boon and/or JQ Chen, be the number of times the class $l$ is generated from the product of classes $j,k$ and $c_j=c_{-j}$ (a ...
2
votes
1answer
142 views

Symmetric subgroups of simple algebraic groups over finite fields

Let $G$ be a simply connected simple algebraic group over a field $k$. Let $\theta\colon G\to G$ be an involution of $G$ over $k$ (an automorphism of order 2). Let $H=(G^\theta)^0$, the identity ...
2
votes
1answer
136 views

$2$-adic valuation on $\mathbb Q (\sqrt{-15})$

I was reading the book "The structure of groups of prime power order". In the book (page 226, Example 10.1.18) it is proved that $-15$ has a square root is $\mathbb Z_2$ where $\mathbb Z_2$ denotes ...
7
votes
0answers
147 views

Cyclic and prime factorizations of finite groups

A tuple $(A_1,\dots,A_n)$ of subsets of a finite group $G$ is called a factorization of $G$ if $G=A_1\cdots A_n$ and $|A_1|\cdots|A_n|=G$. In Cryptology factorizations of groups are known as ...
0
votes
0answers
90 views

Is this basis a Schauder basis?

Let $G$ be a torsion free group. Let $\alpha$ be an element in $\mathbb CG$, the group algebra of $G$, with $\|\alpha\|_1=1$ and assume that $\{1,\alpha,\alpha^2,\dotsc\}$ is linearly independent, $(...
6
votes
0answers
140 views

Hyperbolic group with boundary $S^1$ implies virtually Fuchsian via bounded cohomology?

Question: Is there an approach to $\partial G \cong S^1$ implies virtually Fuchsian using bounded cohomology of $\mathrm{Homeo^+} (S^1)$? If not is there a reason to believe it wouldn't work, or maybe ...
10
votes
3answers
391 views

Is each finite group multifactorizable?

Definition. A finite group $G$ is called multifactorizable if for any positive integer numbers $a_1,\dots,a_n$ with $a_1\cdots a_n=|G|$ there are subsets $A_1,\dots,A_n\subset G$ such that $A_1\cdots ...
12
votes
2answers
498 views

Factorizable groups

Definition. A finite group $G$ is factorizable if for any positive integer numbers $a,b$ with $ab=|G|$ there are subsets $A,B\subset G$ of cardinality $|A|=a$ and $|B|=b$ such that $AB=G$. Problem ...
4
votes
2answers
101 views

A question on UCS p-groups

A $p$-group $G$ is called a ${\it UCS}$ $p$-group if $G$ has precisely three characteristic subgroups, namely $1$, $\Phi(G)$ and $G$. Let $G$ be a finite UCS $p$-group of order $p^{2n}$ such that $\...
1
vote
0answers
100 views

finite groups, class constants relations [closed]

Let $C_{jk}^l$ be the number of times class $l$ is generated from the classes $j$, $k$ and $c_j$ be order of class $j$. See for example finite groups by Jansen and Boon for the notation and some ...
1
vote
0answers
22 views

Defect of subnormality in unit groups of modular group algebras

Let $p$ be a prime number, $G$ a finite p-Group and $K$ a finite field with $char(K)=p$. It is well-known that the group $1+rad(KG)$ is a p-group containing $G$. $G$ is normal in $1+rad(KG)$ if and ...
2
votes
0answers
19 views

Defect of subnormality and repeated normalizer series

Let $G$ be a a finite group and $S$ a subnormal subgroup of $G$. The lenght of a fastest chain of subgroups $(U_i)_{1\le i\le n}$ such that $U_1=S$, $U_i$ normal in $U_{i+1}$ and $U_n=G$ is called the ...
13
votes
2answers
288 views

A finite group that has no decomposition of given cardinality

Let $a,b$ be two positive integer numbers. A group $G$ is called $a{\times}b$-decomposable if there are subsets $A,B\subset G$ of cardinality $|A|=a$ and $|B|=b$ such that $AB=G$ where $AB=\{xy:x\in A,...
7
votes
1answer
284 views

Finite groups containing no subgroups of a given order or index

The classical Lagrange's Theorem says that the order of any subgroup of a finite group divides the order of the group. For abelian groups this theorem can be completed by the following simple fact: ...
7
votes
1answer
107 views

Going up of an amalgamated decomposition of a subgroup of finite index

Let $G$ be a finitely presented group and H a subgroup of index $n$ in $G$. Suppose that H has a non-trivial decomposition as amalgamated product, say $H = A \ast_U B$. I am wondering about the ...
9
votes
2answers
421 views

A question on the fundamental group of a compact orientable surface of genus >1

Let $G=\pi(X,x)$ be the fundamental group of a compact orientable surface of genus $g\ge 2$. It is well known that a presentation of $G$ is $$G=\langle x_1,y_1,\dots,x_g,y_g \ | \ [x_1,y_1]\cdots [x_g,...
7
votes
1answer
224 views

“Almost-ideals” in the (simple) Lie algebra of an algebraic group?

Let $G<\mathrm{GL_n}$ be a simple linear algebraic group defined over a finite field $K$. Let $\mathfrak{g}$ be its Lie algebra. Assume $\mathfrak{g}$ is simple. Is it necessarily the case that ...
3
votes
0answers
100 views

is the functor of finite order elements in grp representable

My question is whether the functor F:Grp->Set that sends a Group to its Set of elements with finite order is representable. I have the sense that it shouldn't be but I've so far failed to prove it in ...
1
vote
0answers
47 views

Relations between Omega, Local Indicable and Right Orderable groups

We know that the set of Right-Orderable groups $RO$, is contained in the set of $\Omega$- groups (Read it from "A Note on Group Rings of Certain Torsion-Free Groups" Burns-Hale). A Group $G$ is a ...
2
votes
1answer
62 views

Centre, FC-centre and finite normal subgroups

The centre of a group $G$ can be described as the set of all elements $g\in G$ whose conjugacy class consists just of $g$ itself. The FC-centre of a group $G$ is the union of all finite conjugacy ...
7
votes
1answer
140 views

Divisors of the regular character of a finite group

Recall that the regular character $\rho=\hspace{-.2cm}\sum\limits_{\chi\in\operatorname{Irr}(G)}\hspace{-.2cm}\chi(1)\chi$ of a finite group $G$ takes values $$ \rho(g)= \left\{\begin{array}{cl} ...
1
vote
1answer
81 views

Does this element belong to all powers of the augmentation ideal of the group algebra.

Let $G$ be a torsion free group, and let $\alpha$ and $\beta$ are elements in the augmentation ideal, $I$, of $\mathbb CG$, the group algebra of $G$. Assume that there exists complex numbers $a$ and $...
6
votes
2answers
382 views

Explicit computation of the Burnside ring

I would like to see explicit computations of the Burnside ring $A(G)$ when $G$ is a small Abelian group, such as $G=\mathbb{Z}/2,\mathbb{Z}/2^n,\mathbb{Z}/p^n$ where $p$ is an odd prime and $n\...
-3
votes
0answers
306 views

Benson Farb conjecture

What does Benson Farb famous open problem say about the Tarski number of the group, precisely?! And what is the imporatnce of it in the study of Tarski number of the groups? I have read something ...
6
votes
1answer
325 views

Action of infinite symmetric groups on iterated power sets

Let $X$ be an infinite set, and $k \ge 1$ be a natural number. We work without the axiom of choice. Let $G_0$ be the full symmetric group on $X$, and let $G_1$ be the full symmetric group on ${\cal ...
3
votes
2answers
151 views

Free ergodic probability measure-preserving actions of the free group

Let $(X,\mathcal{B},\mu)$ be a standard Borel probability space. Let $\Gamma$ be a countable group. An action of $\Gamma$ on $X$ is: essentially free if for all $g \in \Gamma \setminus \{e \}$,...
9
votes
0answers
131 views

Hochschild-Serre spectral sequence via explicit filtration

Let $$1 \longrightarrow K \longrightarrow G \longrightarrow Q \longrightarrow 1$$ be a short exact sequence of groups and let $M$ be a $\mathbb{Z}[G]$-module. The Hochschild--Serre spectral ...
-1
votes
1answer
73 views

Nilpotent subgroups of the direct limit of $GL_n(\mathbb{Z})$ with arbitrarily large finite subgroup

We embeds $GL_n(\mathbb{Z})$ in $GL_{n+1}(\mathbb{Z})$ by identifying $A \in GL_n(\mathbb{Z})$ with $\begin{pmatrix} A & 0 \\ 0 & 1 \end{pmatrix} \in GL_{n+1}(\mathbb{Z})$. Let $GL_\infty(\...
8
votes
2answers
288 views

Hyperbolic $3$-manifold groups that embed in compact Lie groups

Is there a closed hyperbolic $3$-manifold whose fundamental group is isomorphic to a subgroup of some compact Lie group? It is known that every surface group can be embedded into any semisimple ...
4
votes
0answers
104 views

Subgroup membership problem for Noetherian groups

I am interested in the status of the subgroup membership problem (MP) for finitely presented Noetherian groups. That is, given a finite presentation $\langle X,R\rangle$ for a Noetherian group, \begin{...
4
votes
1answer
305 views

Classification of compact connected abelian groups

It is known that torsion-free compact abelian groups are exactly the product of the maximal solenoid group $\Sigma_{(2,3,\cdots)}$ (which is the Pontryagin dual of the additive group $\mathbb{Q}$ of ...
6
votes
2answers
169 views

Generalization of Bieberbach's second theorem

Let $F_0$ and $F_1$ be compact flat manifolds of dimensions $k$ and $m$, respectively, where $k \geq m$. Suppose $f : \pi_1(F_0) \to \pi_1(F_1)$ is a surjective homomorphism. Consider the covering ...
1
vote
0answers
44 views

Which countable discrete groups have a metrisable group compactification?

Let $G$ be a countable discrete group. A group compactification of $G$ is a compact Hausdorff topological group $H$ such that there is a group homomorphism $\iota\colon G\to H$ with dense image. For ...
4
votes
1answer
193 views

Does every index $p$ subgroup of $SL(2,\mathbb{Z}_p)$ contain $\Gamma(p)$?

Does every index $p$ subgroup of $SL(2,\mathbb{Z}_p)$ contain the principal congruence subgroup $\Gamma(p)$? Equivalently, must it be the preimage of an index $p$ subgroup of $SL(2,\mathbb{Z}/p\...
3
votes
0answers
133 views

A permutation group acting on subsets

Consider the the set $$X = \prod_{1 \leq k \leq n-2} \binom{ \bf{n}}{k} $$ where $\binom{ \bf{n}}{k}$ denotes the set of subsets with $k$ elements of the set ${\bf n} = \{1, \cdots , n\}$. For ...
3
votes
1answer
193 views

Fundamental group and group measure space construction

Let $N$ be a type ${\rm II}$ factor, with trace $\tau$. Consider its fundamental group$$ \mathcal{F}(N)= \{ \tau(p)/\tau(q) \ | \ p,q \text{ non-zero finite projections in } N \text{ and } pNp \simeq ...
20
votes
2answers
436 views

$GL_n(\Bbb Z_p)$ conjugacy classes in a $GL_n(\Bbb Q_p)$ conjugacy class

It is easy to classify conjugacy classes in $GL_n(\mathbb Q_p)$ by linear algebra. How to classify $GL_n(\Bbb Z_p)$ conjugacy classes in a $GL_n(\Bbb Q_p)$ conjugacy class? For example, for general ...
2
votes
1answer
97 views

Groups with a maximal subgroup which is solvable

I would like to know results on the structure of a finite group $G$ which possesses a maximal subgroup $H$, with $H$ solvable. More precisely, about supplements of $H$, that is, decompositions $G=HK$ ...
17
votes
2answers
902 views

A character identity

This is related to my question, but it concerns a specific point of the proof of Schur's Theorem. Let $G$ be a finite group and $\chi$ an irreducible character of $G$. Is it true that $$\forall g\in ...
4
votes
1answer
240 views

Finite groups which have trivial outer automorphism group

I was wondering if it is possible to classify the finite groups which have no outer automorphisms? I am currently only aware of the Symmetric Groups ($n \neq 6$) as an infinite class of examples. If ...