Subnormalizers and solvability in finite groups

TL;DR

This paper investigates the probability that a cyclic subgroup generated by an element is subnormal within the subgroup generated by two elements in finite groups, establishing an upper bound for nonsolvable groups.

Contribution

It introduces the invariant sp(G) measuring subnormality probability and proves an upper bound of 1/6 for nonsolvable finite groups.

Findings

sp(G)  1/6 for all nonsolvable groups

Provides a new probabilistic measure related to subgroup properties

Bridges concepts between nilpotent and solvable subgroup generation

Abstract

For a finite group $G$, we study the probability $sp(G)$ that, given two elements $x,y \in G$, the cyclic subgroup $\langle x \rangle$ is subnormal in the subgroup $\langle x, y \rangle$. 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 $sp(G) \leq 1/6$ for every nonsolvable group $G$.