Subnormalizers and solvability in finite groups

For a finite group G, we study the probability sp(G) that, given two elements x, y is an element of G, the cyclic subgroup (x) is subnormal in the subgroup < x, y >. This can be seen as an intermediate invariant between the probability that two elements generate a nilpotent subgroup and the probability that two elements generate a solvable subgroup. We prove that sp(C) <= 1/6 for every nonsolvable group G. Moreover, this bound is best possible. (C) 2022 Elsevier Inc. All rights reserved.