Bases of primitive linear groups.
Let \(G\) be a permutation group on a finite set \(\Omega\) of size \(n\). A subset of \(\Omega\) is said to be a base for \(G\) if its pointwise stabilizer in \(G\) is trivial. The minimal size of a base for \(G\) is denoted by \(b(G)\). A well known conjecture of \textit{L. Pyber} [DIMACS, Ser. Discrete Math. Theor. Comput. Sci. 11, 197-219 (1993; Zbl 0799.20005)] states that there is an absolute constant \(c\), such that if \(G\) is primitive on \(\Omega\), then \(b(G)<c\log|G|/\log n\). The results of the authors [J. Am. Math. Soc. 12, No. 2, 497-520 (1999; Zbl 0916.20003)], \textit{C. Benbenishty} (in preparation) and \textit{Á. Seress} (to appear) reduce the proof of Pyber's conjecture to the case where \(G\) is a primitive affine permutation group, i.e., \(n=p^m\) for some prime \(p\) and \(G=VH\), where \(V\cong(C_p)^m\) is an elementary Abelian subgroup of \(G\), and \(H=G_0\) is a point-stabilizer and an irreducible subgroup of \(\text{GL}(V)\cong\text{GL}_m(p)\). If \(b(H)\) denotes the minimal size of a base for the action of \(H\) on the set of vectors in \(V\), then \(b(G)=1+b(H)\), so the problem reduces to find \(b(H)\) for an irreducible subgroup \(H\) of \(\text{GL}(V)\). The following special cases of the problem have been considered: the case where \(H\) is solvable, see \textit{Á. Seress} [J. Lond. Math. Soc., II. Ser. 53, No. 2, 243-255 (1996; Zbl 0854.20004)] and that in which \(H\) is a \(p'\)-group, see \textit{D. Gluck} and \textit{K. Magaard} [J. Lond. Math. Soc., II. Ser. 58, No. 3, 603-618 (1999; Zbl 0951.20003)]. The main result of this paper is Theorem 1: If \(H\leq\text{GL}(V)\) is an irreducible, primitive linear group on \(V\), then \(b(H)\leq 18\log|G|/\log|V|+c\), where \(c\) is an absolute constant which is explicitly defined. So, Theorem 1 is a strong form of Pyber's conjecture. Theorem 1 follows from a more detailed result, Theorem 2, which describes the structure of irreducible primitive linear groups \(H\) for which \(b(H)\) is unbounded. Theorem 2 may prove useful in an inductive approach to the conjecture for the remaining case of imprimitive linear groups. An interesting intermediate result for the proof of Theorem 2 is Proposition 2.1 in which the authors show that if \(H\) is a quasisimple group of Lie type in characteristic \(p\), and \(V\) is an absolutely irreducible \(H\)-module over a field of characteristic \(p\) of sufficiently large dimension, then \(H\) actually has a regular orbit on \(V\) (i.e., \(b(H)=1\)). Proposition 2.1 refines Theorem 6 from the paper of \textit{J. I. Hall}, the first author and \textit{G. M. Seitz} [Q. J. Math., Oxf. II. Ser. 43, No. 172, 441-458 (1992; Zbl 0780.20020)].
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