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11.80 (1990)

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Let $G$ be a primitive permutation group on a finite set $\Omega$ and suppose that, for $\alpha \in \Omega$, $G_\alpha$ acts 2-transitively on one of its orbits in $\Omega \setminus \{\alpha\}$. By (C. E. Praeger, J. Austral. Math. Soc. (A), 45, 1988, 66–77) either
$\qquad$ (a) $T \leqslant G \leqslant \operatorname{Aut} T$ for some nonabelian simple group $T$, or
$\qquad$ (b) $G$ has a unique minimal normal subgroup which is regular on $\Omega$.
For what classes of simple groups in (a) is a classification feasible? Describe the examples in as explicit a manner as possible. Classify all groups in (b).

Contributor: C. E. Praeger

Progress

Remark of 1999: Two papers by X. G. Fang and C. E. Praeger (Commun. Algebra, 27 (1999), 3727–3754 and 3755–3769) show that this is feasible for $T$ a Suzuki and Ree simple group, and a paper of J. Wang and C. E. Praeger (J. Algebra, 180 (1996), 808–833) suggests that this may not be the case for $T$ an alternating group. Remark of 2001: In (X. G. Fang, G. Havas, J. Wang, European J. Combin., 20, no. 6 (1999), 551–557) new examples are constructed with $G = PSU(3, q)$. Remark of 2009: It was shown in (D. Leemans, J. Algebra, 322, no. 3 (2009), 882–892) that for part (a) classification is feasible for self-paired 2-transitive suborbits with $T$ a sporadic simple group; in this case these suborbits correspond to 2-arc-transitive actions on undirected graphs.

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