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

Abstract

We define and study two arithmetic functions 0 and Á, having domain the set of all finite groups whose orders are not prime powers. Namely, if G is such a group, we call 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 Á.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 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 0 and Á. 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.

On the primary coverings of finite solvable and symmetric groups

“…As a comparison with the previous paper [4], observe the following. Denoting by 0 .G/ the normal primary covering number of a finite group G, that is the smallest natural number such that…”

“…Given a subset … of G, we may be interested in the minimal number of proper subgroups of G whose union contains …. In this paper, we focus on the set of primary elements, complementing the work done in [4]. A primary element of G is an element of G whose order is some prime power.…”

On the Primary Coverings of Finite Solvable and Symmetric Groups

“…Given a subset Π of G we may be interested in the minimal number of proper subgroups of G whose union contains Π. In this paper we focus on the set of primary elements, complementing the work done in [3]. A primary element of G is an element of G whose order is some prime power.…”

“…(2) If n = 2 a for some a > 1, then σ 0 (S n ) = 1 + 1 2 n n/2 (see Proposition 1). (3) If n = 10 and n = 3 ǫ 2 a , for ǫ ∈ {0, 1} and a > 1, then σ 0 (S n ) = 1 + n n2 , where n 2 denotes the maximum power of 2 that divides n (see Proposition 4). (4) If n = 3 • 2 a , with a ≥ 2, then c 1 ≤ σ 0 (S n ) ≤ c 2 , where Proposition 5).…”