Automorphism Groups of the $PSL_2(q)$ Commuting Involution Graphs

TL;DR

The paper determines the automorphism group $A$ of the commuting involution graph $\mathcal{C}(L,X)$ for $L=PSL_2(q)$, distinguishing between even and odd $q$. For odd $q$, the main result is $A\cong P\Gamma L_2(q)$, proven by splitting into $q\equiv 1 \pmod{4}$ and $q\equiv 3 \pmod{4}$ cases and showing that a central involution in a vertex stabilizer must lie in the center for sufficiently large $q$, which forces $L\trianglelefteq A$ and thus $A\le P\Gamma L_2(q)$; together with computational checks for small $q$, this yields the desired equality. The analysis employs the action of $\mathrm{Aut}(L)\cong P\Gamma L_2(q)$ on $\mathcal{C}(L,X)$, detailed control of vertex stabilizers $A_t$, and explicit disc structures around carefully chosen involutions to constrain the possible automorphisms. A key technical component is bounding the number of solutions to certain polynomial equations over finite fields (absolute irreducibility arguments) and translating eigenvalue criteria into graph-distance conditions. The results extend the understanding of automorphism groups of commuting involution graphs in simple groups and relate group-theoretic centralizers to graph symmetries, with small cases verified computationally.

Abstract

Given a finite group $G$ and a conjugacy class of involutions $X$ of $G$, we define the commuting involution graph $\mathcal{C}(G,X)$ to be the graph with vertex set $X$ and $x,y \in X$ adjacent if and only if $x \neq y$ and $xy =yx$. In this paper the automorphism group of the graph $\mathcal{C}(G,X)$ is determined when $G = PSL_2(q)$.

Theorems & Definitions (28)

• Theorem 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Theorem 6