The Number of Solvabilizers in Finite Groups

Abstract

Considering a finite group $G$, for any element $x\in G$, the solvabilizer of $x$ in $G$ is defined as $Sol_G(x)=\{y \in G : \langle x, y \rangle \text{ is solvable}\}$. In this paper, we introduce $Solv(G)$ as the number of distinct solvabilizers of elements in $G$. A group is called $n$-solvabilizer if $|Solv(G)|=n$. We compute $|Solv(G)|$ for various classes of non-abelian simple groups, including $PSL(2, 2^n)$; $PSL(2, 3^n)$ with an odd integer $n$; and $PSL(2, p)$ with a prime $p>7$. Furthermore, we show that for any nonsolvable group $G$, $|Solv(G)|\geq 32$. Finally, we implement an algorithm in GAP for calculating $|Solv(G)|$ for any nonsolvable group $G$. This algorithm can be adapted for all questions generalizing to nilpotent and other subgroup-closed classes of finite groups.