Normal coverings of linear groups

Abstract

For a non-cyclic nite group G, let (G) denote the smallest number of conjugacy classes of proper subgroups of G needed to cover G. Let (G) denote the size of the largest set of conjugacy classes of G, such that any two elements from distinct classes generate G. In this paper several explicit bounds or formulas are given for (G) and (G), where G is a subgroup of GLn(q) containing SLn(q). The results hold also for the group G=Z(G). Motivated by questions in number theory, Bubboloni and Praeger have recently given bounds or exact formulas for (Sn) and (An), for all values of n. Further work of Bubboloni, Praeger and Spiga has established that (Sn) is bounded above and below by linear functions of n. This paper establishes a similar result for linear groups: it is shown that n= 2 2; the upper bound is exact in the case that n is an odd prime.