### A condition for the solvability of finite groups.

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A θ-pair for a maximal subgroup M of a group G is a pair (A, B) of subgroups such that B is a maximal G-invariant subgroup of A with B but not A contained in M. θ-pairs are considered here in some groups having supersoluble maximal subgroups.

The notions of nearly-maximal and near Frattini subgroups considered by J.B. Riles in [20] and the natural related notions are characterized for abelian groups.

Given a class $\mathcal{X}$ of finite groups and a finite group $G$ , the authors study the subgroup $\mathcal{X}\left(G\right)$ intersection of maximal subgroups that do not belong to $\mathcal{X}$.