Groupincluding

Groupincluding is a descriptive term used in informal or expository discussions within abstract algebra to refer to the process of enlarging a group by including additional elements or subgroups, while preserving the original group's structure via an embedding.

Formally, given a group G and a set A of new generators with relations R, a groupincluding of G by A with respect to R consists of a group H and an injective homomorphism i: G → H together with a map from A into H such that H is generated by i(G) together with the images of A, subject to the relations R imposed on those images. A common way to articulate this construction is through a universal property: for any other group K with an injective homomorphism i': G → K and a map from A to K satisfying the same relations, there exists a unique homomorphism φ: H → K with φ ∘ i = i' and φ restricted to i(G) determined by the embedding.

Examples are used to illustrate how a groupincluding operates in practice. If G = Z and A = {r} with the relation r^2 = 1 and r t r^{-1} = t^{-1}, the resulting groupincluding can be viewed as the infinite dihedral group, illustrating how new generators and relations produce a larger group containing the original as a subgroup. In other contexts, groupincluding overlaps with standard constructions such as free products with amalgamation, pushouts in the category of groups, semidirect products, or other group extensions, depending on the chosen relations.

Notes and terminology: groupincluding is not a canonical term in mainstream textbooks; it is a descriptive label for a class of constructions that formalize embedding a given group into a larger one by adjoining generators and relations. See also group extension, amalgamated product, and universal properties in group theory.