On the commuting probability for subgroups of a finite group

Let $K$ be a subgroup of a finite group $G$ . The probability that an element of $G$ commutes with an element of $K$ is denoted by $Pr(K,G)$ . Assume that $Pr(K,G)\geq \epsilon$ for some fixed $\epsilon >0$ . We show that there is a normal subgroup $T\leq G$ and a subgroup $B\leq K$ such that the indices $[G:T]$ and $[K:B]$ and the order of the commutator subgroup $[T,B]$ are $\epsilon$ -bounded. This extends the well-known theorem, due to P. M. Neumann, that covers the case where $K=G$ . We deduce a number of corollaries of this result. A typical application is that if $K$ is the generalized Fitting subgroup $F^{*}(G)$ then $G$ has a class-2-nilpotent normal subgroup $R$ such that both the index $[G:R]$ and the order of the commutator subgroup $[R,R]$ are $\epsilon$ -bounded. In the same spirit we consider the cases where $K$ is a term of the lower central series of $G$ , or a Sylow subgroup, etc.

COMMUTING PROBABILITY OF COMPACT GROUPS

For any (Hausdorff) compact group G, denote by $\mathrm{cp}(G)$ the probability that a randomly chosen pair of elements of G commute. We prove that there exists a finite group H such that $\mathrm{cp}(G)= {\mathrm{cp}(H)}/{|G:F|^2}$ , where F is the FC-centre of G and H is isoclinic to F with $\mathrm{cp}(F)=\mathrm{cp}(H)$ whenever $\mathrm{cp}(G)>0$ . In addition, we prove that a compact group G with $\mathrm{cp}(G)>\tfrac {3}{40}$ is either solvable or isomorphic to $A_5 \times Z(G)$ , where $A_5$ denotes the alternating group of degree five and the centre $Z(G)$ of G contains the identity component of G.

Finite groups with an automorphism inverting, squaring or cubing a non-negligible fraction of elements

Finite groups with an automorphism mapping a sufficiently large proportion of elements to their inverses, squares and cubes have been studied for a long time, and the gist of the results on them is that they are “close to being abelian”. In this paper, we consider finite groups [Formula: see text] such that, for a fixed but arbitrary [Formula: see text], some automorphism of [Formula: see text] maps at least [Formula: see text] many elements of [Formula: see text] to their inverses, squares and cubes. We will relate these conditions to some parameters that measure, intuitively speaking, how far the group [Formula: see text] is from being solvable, nilpotent or abelian; most prominently the commuting probability of [Formula: see text], i.e. the probability that two independently uniformly randomly chosen elements of [Formula: see text] commute. To this end, we will use various counting arguments, the classification of the finite simple groups and some of its consequences, as well as a classical result from character theory.

A generalization of commuting probability of finite rings

The aim of this paper is to study the probability that the commutator of an arbitrarily chosen pair of elements, each from two different additive subgroups of a finite non-commutative ring equals a given element of that ring. We obtain several results on this probability including a computing formula, some bounds and characterizations.

Commutativity preserving extensions of groups

In parallel to the classical theory of central extensions of groups, we develop a version for extensions that preserve commutativity. It is shown that the Bogomolov multiplier is a universal object parametrizing such extensions of a given group. Maximal and minimal extensions are inspected, and a connection with commuting probability is explored. Such considerations produce bounds for the exponent and rank of the Bogomolov multiplier.