HomGrpFS

HomGrpFS is a framework in algebra for the study of homomorphisms between groups equipped with a finite-subgroup structure. In this approach, an object is a pair (G, F) where G is a group and F is a distinguished, nonempty family of subgroups of G that is closed under conjugation and under finite intersections, and often required to satisfy a finiteness condition such as each member of F having finite index in G.

A morphism from (G, F) to (G', F') is a group homomorphism φ: G → G' that preserves the FS-structure: for every H in F, the image φ(H) lies in F'. The hom-set Hom((G, F), (G', F')) consists of all such φ. When F is fixed, HomGrpFS forms a category whose objects and morphisms generalize the usual category of groups by tracking how finite-subgroup data is carried by maps.

This framework is useful for examining questions about how finite-subgroup information propagates through homomorphisms, and for defining invariants that depend on the interaction between subgroups and homomorphisms. It also provides a natural setting for comparing different families F and for studying functorial behavior with respect to restriction and extension of scalars or group actions.

Common choices for F include: (i) the family of all finite subgroups of G, which specializes HomGrpFS to the setting of finitary homomorphisms; and (ii) the family of subgroups of finite index, relevant to profinite and residually finite contexts. In the abelian case, when G and G' are abelian, HomGrpFS reduces to a subcategory of module homomorphisms with additional structure constraints.

Notes: The name HomGrpFS and the exact axioms for F are not universally standardized; different authors may adopt variants tailored to particular problems. See also the category of groups, profinite groups, and the lattice of subgroups.