Subnormalizers and solvability in finite groups

Abstract

For a finite group $G$, we study the probability $sp(G)$ that, given two elements $x,y\in G$, 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 $sp(G)\le 1/6$ for every nonsolvable group $G$.