On the dimension of finite permutation group actions (Q1866600)

The dimension \(\dim(X,G)\) of a finite permutation group \(G\) acting on a set \(X\) is the smallest integer \(r\) with the property that the stabilizer in \(G\) of \(r\) suitable elements is trivial; this \(r\) is the minimal base size of \textit{C. C. Sims} [in Symbolic and algebraic manipulation, Proc. 2nd Symp., Los Angeles 1971, 23-28 (1971; Zbl 0449.20002)], compare also \textit{L. Pyber} [DIMACS, Ser. Discrete Math. Theor. Comput. Sci. 11, 197-219 (1993; Zbl 0799.20005)] and \textit{D. Gluck, Á. Seress} and \textit{A. Shalev} [J. Algebra 199, No. 2, 367-378 (1998; Zbl 0897.20005)]. The author computes this dimension for various examples, like dihedral groups, linear groups, affine groups and projective groups. Furthermore he expresses \(\dim(X,G)\) in terms of the Möbius function of the lattice of subgroups of \(G\), and he investigates the behaviour of this dimension under \(G\)-set constructions (disjoint unions, Cartesian products and wreath products).