Kourovka Notebook Problem 21.111

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.

For a finite nonabelian simple group $S$, put $G_x=\langle x,\operatorname{Inn}(S)\rangle$ for $1\ne x\in\operatorname{Aut}(S)$. Such an $x$ is a $\tau$-automorphism when \[ 3\nmid\left|\langle g^{-1}xg,h^{-1}xh\rangle\right| \qquad\text{for every }g,h\in G_x. \] A $\tau$-group is an $S$ admitting at least one such automorphism. Exclude the Suzuki simple groups ${}^2B_2(q)$, where $q=2^{2m+1}$ with $m\geq1$. The order of an automorphism is its order as an element of $\operatorname{Aut}(S)$.

Statement recorded in the supplied source.

Among finite nonabelian simple groups not isomorphic to any ${}^2B_2(q)$:

(a) classify all $\tau$-groups up to isomorphism;

(b) determine whether a $\tau$-automorphism of odd order exists. The pairwise condition concerns every pair of conjugates in $G_x$, not just a selected generating pair. [S2]

Short English statement

Classify the finite simple groups, apart from Suzuki groups, admitting the source’s pairwise 3-divisibility-avoiding automorphisms, and ask whether any such automorphism has odd order.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2620 (KOU-21.111). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 D. O. Revin and N. Y. Yang, Problem 21.111, 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). https://arxiv.org/abs/1401.0300v48.