On the commuting probability and supersolvability of finite groups

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 Ní Shé 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) .