A short proof of the third Joseph’s conjecture

The commutativity degree d(G) of a finite group G denotes the probability that two randomly chosen elements x,y of G satisfy the equation xy = yx. 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.