A short proof of the third Joseph’s conjecture

The commutativity degree [Formula: see text] of a finite group [Formula: see text] denotes the probability that two randomly chosen elements [Formula: see text] of [Formula: see text] satisfy the equation [Formula: see text]. It has been extensively studied after the classical contributions of Gallagher, Gustafson and Erdős in the sixties. Three relevant conjectures of Joseph appeared from the initial investigations on the topic. Here we offer an alternative (and shorter) argument for the proof of the Third Joseph’s Conjecture, illustrating a series of similar open problems which arise from neighbouring disciplines in recent years.