Orbit invariants and an application to the Baby Monster.
Let \(G\) be a permutation group acting on a set \(X\). A subset \(B\) of \(X\) is called a base for \(G\) if the pointwise stabilizer of \(B\) in \(G\) is trivial. Let \(b(G)\) be the minimal size of a base for a permutation group \(G\). Cameron and Kantor conjectured that there is a constant \(c\) such that \(b(G)\leq c\) for every almost simple group \(G\) in a faithful primitive non-standard action. This conjecture was proved by \textit{M. W. Liebeck} and \textit{A. Shalev}, [J. Am. Math. Soc. 12, No. 2, 497-520 (1999; Zbl 0916.20003)]. Subsequent research showed that \(b(G)\leq 7\) for every finite almost simple group \(G\). In the paper [Isr. J. Math. 177, 307-333 (2010; Zbl 1216.20008)], \textit{T. C. Burness, E. A. O'Brien} and \textit{R. A. Wilson} obtained the precise value of \(b(G)\) for the primitive action of all almost simple sporadic groups \(G\) missing two cases concerning the Baby Monster. Now in the paper under review the authors consider the missing two cases and in addition to the previous result they obtain the precise result as follows: Let \(B\) denote the Baby Monster group and consider it as acting on the right cosets of a maximal subgroup \(H\). If \(H=2^{2+10+20}.(M_{22}:2\times S_3)\), then \(b(B)=3\), and if \(H=2^{30}.L_5(2)\), then \(b(B)=2\).
scientific article; zbMATH DE number 6005438