On the degree of theta pairs of finite groups

Abstract

Let G be a finite group and M a maximal subgroup of G. A θ-pair of M is any pair of subgroups (C,D) of G such that (i) D G, D = G, = M and (iii) C D has no proper normal subgroup of G D . A θ -pair of M is a pair of subgroups (C,D) satisfying conditions (i) and (iii) and a property that D ≤ M and Cg ⊆ M for every g ∈ G. In this paper, we introduce the degree of θ-pairs, denoted by dθ(G) as the ratio |θ(G)|/m(G), where θ(G) is the union of all θ-pairs of the maximal subgroups of G and m(G) is the total number of distinct maximal subgroups of G. Similarly, we define the degrees of maximal θ-pairs, θ -pairs and maximal θ -pairs of a finite group G and give some evaluations on the above degrees for some simple groups, nilpotent groups and solvable groups. Moreover, we prove that if G is nilpotent then the degree of maximal θ-pairs of G is exactly 1.