Related papers: The probability that $x$ and $y$ commute in a comp…

Given a finite group $G$ and a subset $X$ of $G$, the commuting graph of $G$ on $X$, denoted by ${\cal C}(G,X)$, is the graph that has $X$ as its vertex set with $x,y\in X$ joined by an edge whenever $x\neq y$ and $xy=yx$. Let $T$ be a…

The second author introduced notions of weak permutability and commutativity between groups and proved the finiteness of a group generated by two weakly permutable finite groups. Two groups H,K weakly commute provided there exists a…

Let $G$ be a finite group, let $\pi$ be a set of primes and let $p$ be the smallest prime in $\pi$. In this work, we prove that $G$ possesses a normal and abelian Hall $\pi$-subgroup if and only if the probability that two random…

For subsets X,Y of a finite group G, we write Pr(X,Y) for the probability that two random elements x in X and y in Y commute. This paper addresses the relation between the structure of an approximate subgroup A of G and the probabilities…

We prove that for a discrete, countable, and amenable group $G$, if the direct product $G^2=G \times G$ is finitely colored then $\{ g \in G : \text{exists } (x,y) \in G^2 \text{ such that } \{ (x,y),(xg,y),(xg,yg)\} \text{ is…

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…

An automorphism $\alpha$ of a group $G$ is called a commuting automorphism if each element $x$ in $G$ commutes with its image $\alpha(x)$ under $\alpha$. Let $A(G)$ denote the set of all commuting automorphisms of $G$. Rai [Proc. Japan…

Let ${\cal C}$ be a nonempty class of finite groups closed under taking subgroups, homomorphic images and extensions. A subgroup $H$ of an abstract residually ${\cal C}$ group $R$ is said to be conjugacy ${\cal C}$-distinguished if whenever…

If for all $a, b$ in a group $G$, we have that $a^2b^2 = b^2a^2$ and $a^3b^3 = b^3a^3$ then does the group necessarily have to be abelian? This paper shows that the answer is affirmative for finite groups as well as certain classes of…

We extend some properties of random walks on hyperbolic groups to random walks on convergence groups. In particular we prove that if a convergence group $G$ acts on a compact metrizable space $M$ with the convergence property then we can…

A group $G$ is said to be $\frac{3}{2}$-generated if every nontrivial element belongs to a generating pair. It is easy to see that if $G$ has this property then every proper quotient of $G$ is cyclic. In this paper we prove that the…

Every word $w$ in a free group naturally induces a probability measure on every compact group $G$. For example, if $w=\left[x,y\right]$ is the commutator word, a random element sampled by the $w$-measure is given by the commutator…

Let $G$ be a group. Two elements $x,y \in G$ are said to be in the same $z$-class if their centralizers in $G$ are conjugate within $G$. Consider $\mathbb F$ a perfect field of characteristic $\neq 2$, which has a non-trivial Galois…

Let $G$ be an algebraic group. For $d\geq 1$, we define the commuting probabilities $cp_d(G) = \frac{dim(\mathfrak C_d(G))}{dim(G^d)}$, where $\mathfrak C_d(G)$ is the variety of commuting $d$-tuples in $G$. We prove that for a reductive…

For G a finite non-Abelian group we write c(G) for the probability that two randomly chosen elements commute and k(G) for the largest integer such that any k(G)-colouring of G is guaranteed to contain a monochromatic quadruple (x,y,xy,yx)…

We prove that, in a finitely generated residually finite group of subexponential growth, the proportion of commuting pairs is positive if and only if the group is virtually abelian. In particular, this covers the case where the group has…

Let G be a finite group. Define a relation ~ on the conjugacy classes of G by setting C ~ D if there are representatives c \in C and d \in D such that cd = dc. In the case where G has a normal subgroup H such that G/H is cyclic, two…

The solubilizer of an element $x$ of a profinite group $G$ is the set of the elements $y$ of $G$ such that the subgroup of $G$ generated by $x$ and $y$ is prosoluble. We propose the following conjecture: the solubilizer of $x$ in $G$ has…

The subgroup commutativity degree of a group G has been defined in [6] as the probability that two subgroups of G commute, or equivalently that the product of two subgroups is again a subgroup. Problem 4.3 of [6] asks whether there exist…

We study finite groups $G$ with elements $g$ such that $\lvert \mathbf{C}_G(g)\rvert = \lvert G:G' \rvert$. (Such elements generalize fixed-point-free automorphisms of finite groups.) We show that these groups have a unique conjugacy class…