On non self-normalizing subgroups

TL;DR

This paper investigates the structure of finite groups based on the number of conjugacy classes of non-self-normalizing subgroups, revealing solvability and nilpotency properties and classifying groups for small values of n.

Contribution

It characterizes groups with few such conjugacy classes, determines their solvability and nilpotency bounds, and classifies groups in D_n for small n, including specific groups like A_5 and SL_2(3).

Findings

Groups in D_n with n ≤ 3 are solvable with derived length ≤ 2.

A_5 is the unique nonsolvable group in D_4.

Nilpotent groups in D_n have class ≤ n/2 and derived length ≤ log_2(n/2)+1.

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 $n \le 3$, 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 $SL_2(3)$ 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 $\log_2 (n/2) + 1$. 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 $n \ge 3$, then $G$ has derived length at most the minimum of $n-1$ and $3 \log_2 (n+1) + 9$.