Characterizing finite solvable groups through the nilpotency probability

TL;DR

This paper establishes that if the probability of two random elements generating a nilpotent subgroup exceeds 1/12, then the finite group is necessarily solvable.

Contribution

It provides a new probabilistic criterion for solvability of finite groups based on nilpotency probability.

Findings

If nu(G) > 1/12, then G is solvable.

Introduces a probabilistic measure nu(G) for finite groups.

Proves a threshold for nilpotency probability implying solvability.

Abstract

Given a finite group $G$, we denote by $\nu(G)$ the probability that two randomly chosen elements of $G$ generate a nilpotent subgroup. We prove that if $\nu(G)>1/12,$ then $G$ is solvable.