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

We define and study two arithmetic functions, γ 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) the normal primary covering number of G, this is defined as the smallest positive integer k such that the set of primary ele- ments of G is covered by k conjugacy classes of proper (pairwise non-conjugate) subgroups of G. Also we set η(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 and it is much friendlier. The study of these functions for arbitary finite groups reduces im- mediately to the non-abelian simple ones. We therefore apply CFSG to obtain bounds and interesting properties for γ 0 and η. Open questions on these func- tions are reformulated in pure number theoretical terms and lead to problems concerning the distributions and the representations of prime numbers.