Persubgroup

Persubgroup is a term used in abstract algebra to denote a subgroup that remains stable under a specified family of endomorphisms. The phrase is sometimes expanded as “A-invariant subgroup,” where A is a chosen set of endomorphisms of a group G. The concept provides a unified way to describe several classical invariance properties by varying A.

Definition: Let G be a group and A a collection of endomorphisms φ: G → G. A subgroup H ≤ G is a persubgroup of G with respect to A if φ(H) ≤ H for every φ in A. In other words, each endomorphism in A maps H into itself.

Special cases and relations to standard notions:

- If A is the set of all inner automorphisms, a persubgroup is exactly a normal subgroup.

- If A is the full automorphism group Aut(G), the persubgroups are called characteristic subgroups.

- If A consists of all endomorphisms End(G), the persubgroups are the fully invariant subgroups.

Examples: In a finite cyclic group G, endomorphisms act by multiplication, and many subgroups H of G satisfy φ(H) ≤ H for all φ in End(G), making them persubgroups with respect to End(G). In a vector space viewed as an additive group, choosing A to be a set of linear maps yields persubgroups that are invariant subspaces.

Properties: The collection of persubgroups with respect to A forms a sublattice of the subgroup lattice of G, closed under intersections and generated subgroups under appropriate conditions. Extensions and normalizers of persubgroups can be studied by tracking how A interacts with conjugation and inclusion.

Notes: The term persubgroup is not universally standardized. In literature, one may encounter A-invariant subgroup, fully invariant subgroup, or characteristic subgroup depending on the chosen A.

See also: subgroup, normal subgroup, characteristic subgroup, fully invariant subgroup, endomorphism, automorphism, invariant subgroups.