Base sizes for sporadic simple groups

Abstract

Let G be a permutation group acting on a set Ω. A subset of Ω is a base for G if its pointwise stabilizer in G is trivial. We write b(G) for the minimal size of a base for G. We determine the precise value of b(G) for every primitive almost simple sporadic group G, with the exception of two cases involving the Baby Monster group. As a corollary, we deduce that b(G) 7, with equality if and only if G is the Mathieu group M 24 in its natural action on 24 points. This settles a conjecture of Cameron.

Base sizes for simple groups and a conjecture of Cameron

“…For example, suppose that H 0 = 2 G 2 (q 0 ), where q 0 = 3 l and l is odd (note that l 5 since we may assume that q 0 248). Now, if l = 5 then the hypothesis |H| q 32 implies that q 9, and applying [38, Proposition 1.3] we calculate that i 2 (H) < 2(q 0 + 1)q 3 0 < q 11 . Similarly, if l < 5 then i 2 (H) < 3 13 and the desired conclusion quickly follows.…”

“…However, this is strictly an existence result and the proof of [48, Theorem 1.3] does not yield an explicit value for c. Recently, a number of papers have appeared where more explicit base size results are obtained. For example, in the forthcoming paper by T. C. Burness, R. M. Guralnick and J. Saxl, 'Base sizes for actions of simple groups', it is shown that if G has socle A n and n > 12 then b(G) = 2 for all non-standard actions; it quickly follows that b(G) 3 for all n. Minimal base sizes for standard actions of alternating and symmetric groups are determined by James in [29], while precise results for primitive actions of sporadic groups will appear in the forthcoming paper [11]. Non-standard actions of finite classical groups are considered in [7] where it is shown that either b(G) 4, or G = U 6 (2).2, G ω = U 4 (3).2 2 and b(G) = 5.…”

“…Then b(G) 6. Now, the main theorem in [11] states that if G is an almost simple primitive group with sporadic socle then b(G) 7, with equality if and only if G = M 24 acting on 24 points. Therefore, in view of the results discussed above for alternating and classical groups, we see that Theorem 1 completes the proof of Cameron's conjecture in full generality.…”

Extremely primitive sporadic and alternating groups

“…Now, if b(G) = 2, then H has a regular orbit on Ω, so G is not extremely primitive. In [5,6] (see also [10,15]), the primitive almost simple groups G with b(G) = 2 and socle a sporadic or alternating group are determined. This observation greatly simplifies the proof of Theorem 1.1 since we can immediately eliminate the cases with b(G) = 2.…”

“…In addition to the main theorem of Burness, O'Brien and Wilson [6] on base sizes, in our analysis of sporadic groups we make use of the data recorded in the Web Atlas [17] on permutation representations and maximal subgroups of sporadic groups. In many cases, this information allows us to construct G and H as suitable permutation groups (using GAP [9] or Magma [1]) and then quickly determine whether or not G is extremely primitive.…”