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

TL;DR

We address finite groups in $\mathcal{D}_5$, i.e., groups with exactly five conjugacy classes of nontrivial, non self-normalizing subgroups, by leveraging known results for $\mathcal{D}_i$ with $i\le 4$ and the inequality $\mathcal{D}(G/N)\le \mathcal{D}(G)-\mathcal{D}_G(N)-1$. The authors prove that any $G\in\mathcal{D}_5$ is solvable and has derived length $\mathrm{dl}(G)\le 3$, and derive the corollary $\mathrm{dl}(G)\le n-2$ for solvable $G\in\mathcal{D}_n$ with $n\ge 5$. They also exclude the existence of non-solvable or simple groups in $\mathcal{D}_5$, tightening the structural understanding of these families. The results confirm the conjecture of Bianchi et al. and sharpen the general bound on derived length within solvable members of $\mathcal{D}_n$.

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.

Theorems & Definitions (15)

• Theorem A
• Corollary 1
• Lemma 1
• Proposition 1
• Lemma 2
• proof
• Remark 1
• Lemma 3
• proof
• Proposition 2