Abstract

For a group G, a subgroup H of G is said to be conjugate permutable if HH^x = H^xH for any x ∈ G. This concept was introduced by Foguel. In this note, we call a subgroup H self-conjugate permutable in G if HH^x = H^xH implies H^x = H. This is the dual of the concept of conjugate permutable subgroups. A C1-group is a group all of whose cyclic subgroups are self-conjugate permutable; and a C2-group is a group all of whose cyclic subgroups are conjugate permutable or self-conjugate permutable. Some properties of conjugate permutable subgroups and self-conjugate permutable subgroups for a any group are obtained, and the structure of finite C1- and C2-groups are described.