Groupsincluding

Groupsincluding is a term used in informal discussions of group theory to describe the class of groups that contain a given subgroup or family of subgroups. It is not a standard, universally defined concept, but rather a descriptive label that can take slightly different meanings in different texts.

Definition and interpretations

- Fixed subgroup interpretation: For a fixed group H, a group G is said to include H if there exists an embedding φ: H → G with φ(H) as a subgroup of G. In this sense, the class of groups including H consists of all groups in which H can be realized as a substructure.

- Family interpretation: For a family F of groups, a group G includes F if each member of F embeds into G. This captures the idea that G contains copies of all groups in F as subgroups.

Examples

- Including the trivial group: Every group includes the trivial group, so the class including the trivial group is the entire universe of groups.

- Including a cyclic group: If H is cyclic of order n, then a group including H contains an element of order n and thus has a cyclic subgroup of order n.

Notes

- The specific embedding is generally not unique; different embeddings of H into G yield different subgroups of G, even though all witness that G includes H.

- The concept is useful for framing embedding problems, universal constructions, and certain model-theoretic discussions about realizable subgroups within larger groups.

See also: Subgroup, Embedding, Supergroup, Lattice of subgroups, Group inclusion.