The Number of Solvabilizers in Finite Groups

TL;DR

This work introduces the solvabilizer framework for finite groups, defining Sol_G(x) and Solv(G) to count distinct solvabilizers and relate them to group solvability. It establishes a universal lower bound |Solv(G)| ≥ 32 for all nonsolvable G and leverages Thompson’s classification of minimal simple groups to derive exact counts across several families, notably PSL(2, q) and Sz-type groups, with A5 achieving the minimal 32. The paper extends these results to broader simple groups, provides explicit formulas (often piecewise by modulus) for |Solv(PSL(2, q))|, and presents a GAP-based algorithm that computes |Solv(G)| efficiently via rational-class representatives and normalizers. Together, these results illuminate the combinatorial landscape of solvabilizers, enable practical computation, and suggest connections between |Solv(G)| and composition factors such as A5.

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.

Theorems & Definitions (52)

• Lemma 2.2
• Theorem 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• Lemma 2.6
• proof
• Theorem 3.2
• Lemma 3.3
• proof