CAP-subgroups in a direct product of finite groups. (Q858714)

This paper studies the cover-avoidance property in a direct product of groups. If a subgroup \(U\) of a finite group \(G\) has the property that either \(UH=UK\) or \(U\cap H=U\cap K\) for every chief factor \(H/K\) of \(G\), then \(U\) is said to have the cover-avoidance property in \(G\) and is called a CAP-subgroup of \(G\). It is well-known that a subgroup \(U\) of a direct product \(G_1\times G_2\) is determined by isomorphic sections \(S_1\) of \(G_1\) and \(S_2\) of \(G_2\) and by an isomorphism \(\varphi\) between those sections. The author proves that whether \(U\) is a CAP-subgroup of \(G_1\times G_2\) depends on the isomorphism \(\varphi\), but not necessarily on the sections \(S_1\) and \(S_2\). Equivalently, \(U\) is a CAP-subgroup of \(G_1\times G_2\) if and only if \(UM\cap G_1\) is a CAP-subgroup of \(G_1\) and \(UN\cap G_2\) is a CAP-subgroup of \(G_2\) for all \(M\trianglelefteq G_2\) and \(N\trianglelefteq G_1\). Consequently, subdirect subgroups and CAP-subgroups of direct factors have the cover-avoidance property.