CAP-subgroups in a direct product of finite groups

If a subgroup U of a finite group G has the property that either UH=UK or U∩H=U∩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 G1×G2 is determined by isomorphic sections S1 of G1 and S2 of G2 and by an isomorphism ϕ between those sections. We prove that whether U is a CAP-subgroup of G1×G2 depends on the isomorphism ϕ, but not necessarily on the sections S1 and S2. Equivalently, U is a CAP-subgroup of G1×G2 if and only if UM∩G1 is a CAP-subgroup of G1 and UN∩G2 is a CAP-subgroup of G2 for all and . Consequently, subdirect subgroups and CAP-subgroups of direct factors have the cover-avoidance property.