Supergroups

Supergroups are a concept used in multiple domains, most prominently in mathematics and music. In group theory, a supergroup of a subgroup H is any group G that contains H as a subgroup. If H ≤ G, then G is a supergroup of H. This idea helps researchers study how a subgroup sits inside larger structures, track which properties of H persist in bigger groups, and understand how different groups relate to one another. There is no unique or canonical supergroup of H; many groups can contain H as a subgroup.

In the context of a fixed ambient group, several related notions clarify the structure of supergroups. The normalizer N_G(H) consists of all elements g in G that conjugate H to itself (gHg^{-1} = H). H is normal in N_G(H), and N_G(H) is the largest subgroup of G in which H is normal. If H is not normal in G, G remains a supergroup of H without a guaranteed quotient structure. The smallest supergroup of H containing a given set of elements S is the subgroup generated by H together with S, denoted ⟨H ∪ S⟩.

Outside mathematics, the term supergroup also describes a musical ensemble formed by members who are already famous from other acts. Such bands are typically project-oriented and may span different genres. Notable examples include Cream and Asia, which brought together established artists for joint projects. In more advanced mathematics, “supergroup” can refer to Lie supergroups or algebraic supergroups, where the group structure is extended to include graded components consistent with supersymmetry concepts.

Overall, supergroups describe how a given subgroup sits inside larger algebraic or collaborative structures, with both formal and cultural uses across disciplines.