Hypothengruppe

Hypothengruppe, or hypothesis group, is an informal term used in some German-language mathematical writings to denote a subgroup of a group defined by a specified set of hypotheses about its elements or about the group's action. It is not a standard, widely adopted notion in formal group theory.

Definition: Let G be a group and H a set of predicates or conditions on elements of G. Define S(H) = { g in G | g satisfies every condition in H }. The Hypothengruppe associated with G and H is Hyp(G,H) = ⟨S(H)⟩, the subgroup generated by all elements that satisfy H. If S(H) is empty, Hyp(G,H) is the trivial subgroup. Variants may define Hyp(G,H) as the largest subgroup of G in which all elements satisfy H, in which case one takes Hyp(G,H) = { g in G | g satisfies H } when that set is a subgroup; otherwise the generated subgroup is taken.

Examples: In a group acting on a set, H might specify properties preserved by the action (for instance, fixing a subset or satisfying certain relations). The associated Hypothengruppe then captures the portion of G that enforces or respects those hypotheses.

Properties: If H1 ⊆ H2, then S(H1) ⊆ S(H2), and hence Hyp(G,H1) ⊆ Hyp(G,H2). If the defining hypotheses are invariant under conjugation by G, the resulting subgroup may be normal. In general, the construction is not a standard one, and its algebraic properties depend on the nature of the hypotheses.

Usage: The term appears mainly in pedagogical contexts or historical discussions. In contemporary practice, one would typically refer to a subgroup defined by a family of conditions, a stabilizer, or the subgroup generated by a given subset.

See also: Subgroup, Generator, Stabilizer, Normal subgroup, Group action.