21.52 (2026)

Let $L$ be a finite non-abelian simple group, and let $D$ be a conjugacy class of involutions in $L$. Consider the complete graph $\Gamma$ with vertex set $D$. Define an equivalence relation $\sim$ (graph coloring) on the set of edges as follows: $(a, b) \sim (c, d)$ if and only if $|ab| = |cd|$. An automorphism of the coloured graph $\Gamma$ is a permutation $\tau \in S_D$ such that $(a, b) \sim (a^\tau, b^\tau)$ for every edge $(a, b)$. Is it true that the automorphism group of $\Gamma$ is a subgroup of $\text{Aut}(L)$?