Groups with 5 nontrivial conjugacy classes of non self-normalizing subgroups are solvable

TL;DR

This paper verifies a conjecture that groups with exactly five conjugacy classes of nontrivial, non self-normalizing subgroups are solvable with derived length at most three.

Contribution

It confirms the conjecture that all groups in the class _5 are solvable with derived length at most three, advancing understanding of subgroup conjugacy class structures.

Findings

Confirmed the conjecture for _5 groups

All such groups are solvable

Derived length is at most three

Abstract

For any $n$ nonnegative integer a family of groups, denoted by $ \mathcal{D}_n $, was introduce by Bianchi et al., as the collection of all finite groups with exactly $n$ conjugacy classes of nontrivial, non self-normalizing subgroups. It was conjectured that $\mathcal{D}_5$ consists of solvable groups with derived length at most $3$. In this note we verify their conjecture.