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18.115 (2014)

Solved

Let $G$ be a finite simple group, and let $X, Y$ be isomorphic simple maximal subgroups of $G$. Are $X$ and $Y$ conjugate in $\text{Aut } G$?

Contributor: P. Schmid

Progress

No, not always. For example, there are two $\text{Aut } G$-classes of $M_{12}$ in $E_6(5)$ (P. B. Kleidman, R. A. Wilson, J. London Math. Soc., 42 (1990), 555–561); other counterexamples in $E_6(q)$ for certain $q$ include two classes of $PSL_2(11)$, and two of $PSL_2(19)$ (D. A. Craven, Invent. Math., 234, no. 2 (2023), 637–719). (D. A. Craven Letter of 8 September 2021.)

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