Extremely primitive sporadic and alternating groups

Abstract

A non-regular primitive permutation group is said to be extremely primitive if a point stabilizer acts primitively on each of its orbits. By a theorem of Mann and the second and third authors, every finite extremely primitive group is either almost simple or of affine type. In a recent paper, we classified the extremely primitive almost simple classical groups, and in this note we determine the examples with a sporadic or alternating socle. We obtain two infinite families for A n (or Sn); they comprise the natural 2-primitive action of n points, plus the action on partitions of {1, . . . , n} into subsets of size n/2 (with n/2 odd). There are twenty examples for sporadic groups, including the rank 6 representation of Co 2 on the cosets of McL.

A note on extremely primitive affine groups

“…For example, the natural actions of Sym n and PGL 2 (q) of degree n and q +1, respectively, are extremely primitive. The study of these groups can be traced back to work of Manning [19] in the 1920s and they have been the subject of several papers in recent years [6][7][8]18].…”

“…A key theorem of Mann, Praeger, and Seress [18, Theorem 1.1] states that every extremely primitive group is either almost simple or affine, and in the same paper they classify all the affine examples up to the possibility of finitely many exceptions. In later work, Burness, Praeger, and Seress [6,7] determined all the almost simple extremely primitive groups with socle an alternating, classical, or sporadic group. The classification for almost simple groups has very recently been completed in [8], where the remaining exceptional groups of Lie type are handled.…”

Extremely primitive groups and linear spaces

“…Combining Theorems 1.1 and 1.2 with [5], Theorem 1, and [4], Theorem 1.1, we have the following corollary. Corollary 1.4.…”

“…The extremely primitive permutation groups of almost simple type are classified in [5], [4], except the case of exceptional groups.…”

“…According to [15], Theorem 1.1, a finite extremely primitive group is either of affine type or almost simple. The almost simple extremely primitive groups whose socle is sporadic, alternating or classical are classified by Burness, Praeger and Seress [5], [4]. In this paper, we use these works to investigate the finite linear spaces admitting an extremely primitive automorphism group.…”