groupsuch

Groupsuch is a mathematical concept defined as a pair (G, S), where G is a group and S is a nonempty family of subgroups of G, called the group-such. The defining properties are: S is closed under conjugation by elements of G; S is closed under finite joins, so for any H1,...,Hk in S the subgroup generated by their union ⟨H1 ∪ ... ∪ Hk⟩ lies in S; and the union of all members of S equals G with the intersection of all members being {e}. When these hold, (G, S) is a groupsuch.

Properties: Because S covers G and is closed under conjugation and joins, every element of G lies in some member of S, and the subgroup lattice of G is reflected by S. The structure enables a modular view of G, and morphisms that preserve S send groupsuch structures to groupsuch structures, preserving the distinguished subgroups under appropriate maps.

Examples: A finite cyclic group G of order n with S equal to the full subgroups lattice of G is a groupsuch; all subgroups are unique for each divisor and are normal, the joins stay within S, and the intersection of all subgroups in S is {e}. Any abelian group G with S taken as the set of all subgroups is also a groupsuch, since conjugation is trivial and joins of subgroups remain subgroups in S.

Relation and uses: Groupsuch connects subgroup lattice theory with group decomposition approaches in algebraic combinatorics and model theory. It provides a framework for comparing groups via their distinguished subgroup families and for studying how global properties of G relate to the local data contained in S. See also subgroup lattice, group action, and lattice theory.