(SUNY at Binghamton)
A subgroup H of a group G is supplemented in G if there is a subgroup K of G such that G = HK. We investigate groups in which all subgroups, all normal subgroups, or all characteristic subgroups have a proper supplement. This supplement can be either an arbitrary subgroup, a normal or a characteristic subgroup, resulting in nine classes of groups. Properties of these classes are studied such as containment and closure properties, and characterizations for several of these classes are given.