Limit points in the range of the commuting probability function on finite groups

TL;DR

This paper investigates the limit points of the commuting probability function in finite groups, showing they are rational and establishing gaps near these points, thus supporting longstanding conjectures.

Contribution

It proves that limit points of the commuting probability are rational and identifies gaps near these points, advancing understanding of the structure of finite groups.

Findings

Limit points of Pr(G) are rational numbers.

Existence of gaps near these limit points.

Supports conjectures of Keith Joseph.

Abstract

If G is a finite group, then Pr(G) denotes the fraction of ordered pairs of elements of G which commute. We show that, if l \in (2/9,1] is a limit point of the function Pr on finite groups, then l \in \Q and there exists an e = e_l > 0 such that Pr(G) \not\in (l - e_l, l) for any finite group G. These results lend support to some old conjectures of Keith Joseph.