Kourovka Notebook Problem 21.53

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $G$ be a finite nonabelian simple group and $D$ a conjugacy class of involutions (elements of order $2$). Let $\Gamma$ be the complete graph with vertex set $D$, whose edge $\{a,b\}$ is coloured by the order $|ab|$ of the group element $ab$. An automorphism of this coloured graph preserves each colour. Write \[ \operatorname{Aut}_t(\Gamma)=\{\tau\in\operatorname{Sym}(D): |\tau(a)\tau(b)|=t\text{ whenever }a,b\in D,\ |ab|=t\}. \] Because $D$ is finite, this one-way preservation of a given colour is equivalent to preserving its colour class as a set, and $\operatorname{Aut}(\Gamma)=\bigcap_t\operatorname{Aut}_t(\Gamma)$. Let $p$ be the second-smallest prime divisor of $|G|$, the smallest being $2$.

Statement recorded in the supplied source.

For every such group $G$ and class $D$, is \[ \operatorname{Aut}(\Gamma)=\operatorname{Aut}_2(\Gamma)\cap\operatorname{Aut}_p(\Gamma)? \] Thus, does preserving the two edge colours given by the smallest prime divisors force preservation of every product-order colour? [S2]

Short English statement

For a conjugacy class of involutions in a finite simple group, do the first two prime product orders determine every colour-preserving symmetry?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2562 (KOU-21.53). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 I. B. Gorshkov, Problem 21.53, in E. I. Khukhro and V. D. Mazurov (eds.), The Kourovka Notebook: Unsolved Problems in Group Theory, 21st edition, arXiv:1401.0300v48 (6 October 2026). See also source definitions in Problems 21.52. This version records a negative answer; the displayed question is the historically registered target. https://arxiv.org/abs/1401.0300v48.