Generalizing the Hilton–Mislin Genus Group

For any group H, let χ(H) be the set of all isomorphism classes of groups K such that K × H × . For a finitely generated group H having finite commutator subgroup [H, H], we define a group structure on χ(H) in terms of embeddings of K into H, for groups K of which the isomorphism classes belong to χ(H). If H is nilpotent, then the group we obtain coincides with the genus group (H) defined by Hilton and Mislin. We obtain some new results on Hilton–Mislin genus groups as well as generalizations of known results.