On the commuting probability and supersolvability of finite groups

Abstract

For a finite group $$G$$ , let $$d(G)$$ denote the probability that a randomly chosen pair of elements of $$G$$ commute. We prove that if $$d(G)>1/s$$ for some integer $$s>1$$ and $$G$$ splits over an abelian normal nontrivial subgroup $$N$$ , then $$G$$ has a nontrivial conjugacy class inside $$N$$ of size at most $$s-1$$ . We also extend two results of Barry, MacHale, and Ni She on the commuting probability in connection with supersolvability of finite groups. In particular, we prove that if $$d(G)>5/16$$ then either $$G$$ is supersolvable, or $$G$$ isoclinic to $$A_4$$ , or $$G/\mathbf{Z}(G)$$ is isoclinic to $$A_4$$ .