Related papers: Twisted Conjugacy in Houghton's groups

In this paper, we show that each finite group $G$ containing at most $p^2$ Sylow $p$-subgroups for each odd prime number $p$, is a solvable group. In fact, we give a positive answer to the conjecture in \cite{Rob}.

In the 1980's K.S. Brown proved that the Houghton group $H_n$ is of type $\operatorname{F}_{n-1}$ but not $\operatorname{FP}_n$. We show that, provided $n\ge3$, the same conclusion holds for all subgroups $G$ of $H_n$ that are 'large' in…

We construct a class of finitely generated groups which have arbitrarily large conjugacy separability function, but in which the conjugacy problem can be solved in polynomial time, demonstrating that the McKinsey algorithm for the conjugacy…

Let $G$ be a linear algebraic group over an algebraically closed field of characteristic $p\geq 0$. We show that if $H_1$ and $H_2$ are connected subgroups of $G$ such that $H_1$ and $H_2$ have a common maximal unipotent subgroup and…

We prove that a finitely generated solvable group which is not virtually nilpotent has exponential conjugacy growth.

We study the algebraic structure and representation theory of the Hopf algebras ${}_J\mathcal{O}(G)_J$ when $G$ is an affine algebraic unipotent group over $\mathbb{C}$ with $\mathrm{dim}(G) = n$ and $J$ is a Hopf $2$-cocycle for $G$. The…

We address two variants of the classical necklace counting problem from enumerative combinatorics. In both cases, we fix a finite group $\mathcal{G}$ and a positive integer $n$. In the first variant, we count the ``identity-product…

Let $S$ be a semigroup. The elements $a,b\in S$ are called primarily conjugate if $a=xy$ and $b=yx$ for certain $x,y\in S$. The relation of conjugacy is defined as the transitive closure of the relation of primary conjugacy. In the case…

The transitive simultaneous conjugacy problem asks whether there exists a permutation $\tau \in S_n$ such that $b_j = \tau^{-1} a_j \tau$ holds for all $j = 1,2, \ldots, d$, where $a_1, a_2, \ldots, a_d$ and $b_1, b_2, \ldots, b_d$ are…

It is well known that if $G$ admits a f.g. subgroup $H$ with a weaklyaperiodic SFT (resp. an undecidable domino problem), then $G$itself has a weakly aperiodic SFT (resp. an undecidable domino problem).We prove that we can replace the…

Let $G$ be a finite group and $N<G$ a normal subgroup with $G/N$ abelian. We show how the conjugacy classes of $G$ in a given coset $qN$ relate to the irreducible characters of $G$ that are not identically $0$ on $qN$. We describe several…

We introduce new obstructions to topological knot concordance. These are obtained from amenable groups in Strebel's class, possibly with torsion, using a recently suggested $L^2$-theoretic method due to Orr and the author. Concerning…

Let $D_n \subseteq \Sigma_n$ be the set of derangements in the symmetric group. We prove that the complex of injective words generated by $\Sigma_n \setminus D_n$ is contractible. This gives a conceptual explanation to the well known fact…

Let $H$ be a nonabelian finite simple group. Huppert's conjecture asserts that if $G$ is a finite group with the same set of complex character degrees as $H$, then $G\cong H\times A$ for some abelian group $A$. Over the past two decades,…

Suppose, $G$ is a residually finite group of finite upper rank admitting an automorphism $\varphi$ with finite Reidemeister number $R(\varphi)$ (the number of $\varphi$-twisted conjugacy classes). We prove that such $G$ is soluble-by-finite…

We show that the word problem of the Brin-Higman-Thompson group $n G_{k,1}$ is {\sf coNP}-complete for all $n \ge 2$ and all $k \ge 2$. For this we prove that $n G_{k,1}$ is finitely generated, and that $n G_{k,1}$ contains a subgroup of $2…

Say that a finite group $G$ is mixable if a product of random elements, each chosen independently from two options, can distribute uniformly on $G$. We present conditions and obstructions to mixability. We show that $2$-groups, the…

Let $H$ be an extension of a finite group $Q$ by a finite group $G$. Inspired by the results of duality theorems for \'etale gerbes on orbifolds, we describe the number of conjugacy classes of $H$ that maps to the same conjugacy class of…

Let $\mathcal{M}(n)$ be the subgroup of $GL(n,\mathbb{Z})$ generated by the particular involutions that are identical to the identity, except for a single line where $-1$ and $+1$ alternate. We study the properties of $\mathcal{M}(n)$, and…

The braid group has recently attracted much attention. This is primarily based upon the discovery of its usage in various cryptosystems [AAG],[KLCHKP]. One major focus of current research has been in solving decision problems in braid…