ZweiUntergruppierung

ZweiUntergruppierung is a term found in some German-language expositions of group theory to denote the operation of combining two subgroups of a group into a single subgroup that reflects their interaction. In a group G with subgroups H and K, the zweiUntergruppierung of H and K typically refers to the subgroup generated by H and K, denoted ⟨H ∪ K⟩. This is the smallest subgroup of G that contains both H and K and is also known as the join of H and K in the lattice of subgroups.

If the product HK satisfies HK = KH, then HK is itself a subgroup and coincides with ⟨H ∪ K⟩. In general, however, HK need not be a subgroup, and ⟨H ∪ K⟩ always provides the subgroup generated by the two given subgroups.

Properties commonly noted include: monotonicity with respect to enlarging H or K, and associativity in the sense that the join of three subgroups satisfies ⟨H ∪ K ∪ L⟩, which is the same as the join of ⟨H ∪ K⟩ with L. In abelian groups, ⟨H ∪ K⟩ equals the product HK, since all subgroups commute.

Examples illustrate the concept: in the symmetric group S3, H = ⟨(12)⟩ and K = ⟨(23)⟩ generate the whole group, so ⟨H ∪ K⟩ = S3.

See also: subgroup, join, product of subgroups, lattice of subgroups. References: standard texts on group theory.