CAP-subgroups in a direct product of finite groups

If a subgroup U of a finite group G has the property that either U H = UK or U boolean AND H = U boolean AND 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) x G(2) is determined by isomorphic sections S-1 of G(1) and S-2 of G(2) and by an isomorphism phi between those sections. We prove that whether U is a CAP-subgroup of G(1) x G(2) depends on the isomorphism phi, but not necessarily on the sections S-1 and S-2. Equivalently, U is a CAP-subgroup of G(1) x G(2) if and only if U M boolean AND G(1) is a CAP-subgroup of G(1) and U N boolean AND G(2) is a CAP-subgroup of G(2) for all M <= G(2) and N <= G(1). Consequently, subdirect subgroups and CAP-subgroups of direct factors have the cover-avoidance property. (c) 2006 Elsevier Inc. All rights reserved.