Base sizes for sporadic simple groups

Abstract

Let G be a permutation group acting on a set
. A subset of
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) 6 7, with equality if and only if G is the Mathieu group
M24 in its natural action on 24 points. This settles a conjecture of Cameron.

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