# Questions tagged [gr.group-theory]

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

**7**

votes

**1**answer

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

**2**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**3**answers

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

**2**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**2**answers

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

**1**answer

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 ...