Related papers: Conjugacy in Houghton's Groups

We classify surface Houghton groups, as well as their pure subgroups, up to isomorphism, commensurability, and quasi-isometry.

Let $G$ be a finite group and $N(G)$ be the set of its conjugacy class sizes. In the 1980's Thompson conjectured that the equality $N(G)=N(S)$, where $Z(G)=1$ and $S$ is simple, implies the isomorphism $G\simeq S$. In a series of papers of…

Determining the length of short conjugators in a group can be considered as an effective version of the conjugacy problem. The conjugacy length function provides a measure for these lengths. We study the behaviour of conjugacy length…

For a Coxeter group (W,S), a permutation of the set S is called a Coxeter word and the group element represented by the product is called a Coxeter element. Moving the first letter to the end of the word is called a rotation and two Coxeter…

We consider pairs of finitely presented, residually finite groups $P\hookrightarrow\G$ for which the induced map of profinite completions $\hat P\to \hat\G$ is an isomorphism. We prove that there is no algorithm that, given an arbitrary…

We show that for $n \ge 6$ every even permutation on $n$ symbols is the commutator of two $n$-cycles. More precisely, let $S_n$ be the symmetric group and $A_n$ the alternating group. Let $C(n) \subset S_n$ denote the conjugacy class of…

This paper is devoted to the theory of $GL_n({\mathbb Z})$-conjugacy classes of regular integer $n\times n$ matrices. Such a matrix is $GL_n({\mathbb Q})$-conjugate to the companion matrix of its characteristic polynomial. But the set of…

Finite hamiltonian groups are counted. The sequence of numbers of all groups of order $n$ all whose subgroups are normal and the sequence of numbers of all groups of order less or equal to $n$ all whose subgroups are normal are presented.

We prove that finite index subgroups of right angled Artin groups are conjugacy separable. We then apply this result to establish various properties of other classes of groups. In particular, we show that any word hyperbolic Coxeter group…

We study a family of groups consisting of the simplest extensions of lamplighter groups. We use these groups to answer multiple open questions in combinatorial group theory, providing groups that exhibit various combinations of properties:…

A question of interest both in Hopf-Galois theory and in the theory of skew braces is whether the holomorph $\mathrm{Hol(N)}$ of a finite soluble group $N$ can contain an insoluble regular subgroup. We investigate the more general problem…

Let $M$ be a model set meeting two simple conditions: (1) the internal space $H$ is a product of $R^n$ and a finite group, and (2) the window $W$ is a finite union of disjoint polyhedra. Then any point pattern with finite local complexity…

Let $G$ be a finite group. Let $U_1,U_2,\dots$ be a sequence of orthogonal representations in which any irreducible representation of $\oplus_{n \geq 1} U_n$ has infinite multiplicity. Let $V_n=\oplus_{i=1}^n U_n$ and $S(V_n)$ denote the…

According to T. Foguel a subgroup $H$ of a group $G$ is called conjugate-permutable if $ HH^x=H^xH$ for every $x\in G$. Mingyao Xu and Qinhai Zhang studied finite groups with every subgroup conjugate-permutable (ECP-groups) and asked three…

Given a group $G$, we write $x^G$ for the conjugacy class of $G$ containing the element $x$. A famous theorem of B. H. Neumann states that if $G$ is a group in which all conjugacy classes are finite with bounded size, then the derived group…

A Young subgroup of the symmetric group $\mathcal{S}_{N}$, the permutation group of $\{ 1,2,\dots,N\} $, is generated by a subset of the adjacenttranspositions $\{ ( i,i+1) \mid 1\leq i < N\}$. Such a group is realized as the stabilizer…

Let $\mathbb Z_n$ denote the cyclic group of order $n$. We show how the group determinant for $G= \mathbb Z_n \times H$ can be simply written in terms of the group determinant for $H$. We use this to get a complete description of the…

Let G be a word-hyperbolic group with given finite generating set, for which various standard structures and constants have been pre-computed. A (non-practical) algorithm is described that, given as input two lists A and B, each composed of…

In this paper we study the complexity of solving quadratic equations in the lamplighter group. We give a complete classification of cases (depending on genus and other characteristics of a given equation) when the problem is…

The permutation matrices form a subgroup of $\text{GL}_n(\mathbb{C})$ that is isomorphic to the symmetric group $S_n$. Let $r_{\mu\lambda}$ denote the multiplicity of the irreducible representation $V_\mu$ of $S_n$, corresponding to a…