The embedding of CAP-subgroups in finite groups

This paper features new and unpublished results about the cover-avoidance property presented in the context of previously established ones. The goal of this work is to contribute to the breadth of the current body of knowledge and to fill in some gaps in the literature. On one hand, we summarize and investigate further the relationships between the cover-avoidance property and normal, conjugate, Sylow, and maximal subgroups. On the other hand, we define a subgroup U to be a subCAP-subgroup of the finite group G if there exists a chain of subgroups \({U = U_{0}\leq U_{1}\leq \dotsm \leq U_{r} = G}\) such that U _i-1 is a CAP-subgroup of U _i for all \({i \in \{1, \dots, r\}}\), and show how this property can be used to characterize solvable groups.