On the common transversal probability in finite groups

TL;DR

This paper investigates the probability that a randomly chosen left transversal of a subgroup in a finite group is also a right transversal, defining a new invariant that reveals structural properties of the group.

Contribution

It introduces the concept of common transversal probability, analyzes its bounds and values, and links it to the structural characteristics of finite groups.

Findings

Bounds for the invariant $ ext{tp}(G)$ are established.

$ ext{tp}(G)$ can determine certain structural properties of $G$.

Several open problems related to transversal probabilities are discussed.

Abstract

Let $G$ be a finite group, and let $H$ be a subgroup of $G$. We compute the probability, denoted by $P_G(H)$, that a left transversal of $H$ in $G$ is also a right transversal, thus a two-sided one. Moreover, we define, and denote by $\mathrm{tp}(G)$, the common transversal probability of $G$ to be the minimum, taken over all subgroups $H$ of $G$, of $P_G(H)$. We prove a number of results regarding the invariant $\mathrm{tp}(G)$, like lower and upper bounds, and possible values it can attain. We also show that $\mathrm{tp}(G)$ determines structural properties of $G$. Finally, several open problems are formulated and discussed.