Kourovka Notebook Problem 21.110

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 $S$, identify $\operatorname{Inn}(S)$ with its inner automorphism group inside $\operatorname{Aut}(S)$. For $1\ne x\in\operatorname{Aut}(S)$ put $G_x=\langle x,\operatorname{Inn}(S)\rangle$ and \[ \alpha_S(x)=\min\{m:\exists g_1,\ldots,g_m\in G_x, \langle g_1^{-1}xg_1,\ldots,g_m^{-1}xg_m\rangle=G_x\}. \] Conjugation is within $G_x$, not within the full automorphism group. Exceptional finite groups of Lie type are the standard simple groups from the exceptional root systems $G_2,F_4,E_6,E_7,E_8$ and their twisted analogues, with parameters for which the group is simple.

Statement recorded in the supplied source.

(a) If $S$ is an exceptional finite simple group of Lie type, must $\alpha_S(x)\leq5$ for every nonidentity $x\in\operatorname{Aut}(S)$?

(b) For each such $S$, determine the exact maximum \[ \max_{x\in\operatorname{Aut}(S)\setminus\{1\}}\alpha_S(x). \] Both inner and outer automorphisms are included. [S2]

Short English statement

For each exceptional finite simple group of Lie type, bound by five the conjugates needed to generate with an automorphism, and determine the exact worst-case number.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2619 (KOU-21.110). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. Guralnick, J. Saxl and D. O. Revin, Problem 21.110, 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.