On the indices of maximal subgroups and the normal primary coverings of finite groups

Abstract

Abstract We define and study two arithmetic functions γ 0 {\gamma_{0}} and η, having domain the set of all finite groups whose orders are not prime powers. Namely, if G is such a group, we call γ 0 ⁢ ( G ) {\gamma_{0}(G)} the normal primary covering number of G; this is defined as the smallest positive integer k such that the set of primary elements of G is covered by k conjugacy classes of proper (pairwise non-conjugate) subgroups of G. Also we set η ⁢ ( G ) {\eta(G)} , the indices covering number of G, to be the smallest positive integer h such that G has h proper subgroups having coprime indices. This second function is an upper bound for γ 0 {\gamma_{0}} , and it is much friendlier. The study of these functions for arbitrary finite groups reduces immediately to the non-abelian simple ones. We therefore apply CFSG to obtain bounds and interesting properties for γ 0 {\gamma_{0}} and η. Open questions on these functions are reformulated in pure number-theoretical terms and lead to problems concerning the distributions and the representations of prime numbers.