On non self-normalizing subgroups

Mariagrazia Bianchi; Rachel D. Camina; Mark L. Lewis; Emanuele Pacifici; Lucia Sanus

paper

On non self-normalizing subgroups

arXiv:2411.18102

Abstract

Let $n$ be a non negative integer, and define $D_{n}$ to be the family of all finite groups having precisely $n$ conjugacy classes of nontrivial subgroups that are not self-normalizing. We are interested in studying the behavior of $D_{n}$ and its interplay with solvability and nilpotency. We first show that if $G$ belongs to $D_{n}$ with $n \leq 3$ , then $G$ is solvable of derived length at most 2. We also show that $A_{5}$ is the unique nonsolvable group in $D_{4}$ , and that $S L_{2} (3)$ is the unique solvable group in $D_{4}$ whose derived length is larger than 2. For a group $G$ , we define $D (G)$ to be the number of conjugacy classes of nontrivial subgroups that are not self-normalizing. We determine the relationship between $D (H \times K)$ and $D (H)$ and $D (K)$ . We show that if $G$ is nilpotent and lies in $D_{n}$ , then $G$ has nilpotency class at most $n /2$ and its derived length is at most $lo g_{2} (n /2) + 1$ . We consider $D_{n}$ for several classes of Frobenius groups, and we use this classification to classify the groups in $D_{0}$ , $D_{1}$ , $D_{2}$ , and $D_{3}$ . Finally, we show that if $G$ is solvable and lies in $D_{n}$ with $n \geq 3$ , then $G$ has derived length at most the minimum of $n - 1$ and $3 lo g_{2} (n + 1) + 9$ .