singlegroup

In mathematics, singlegroup is an informal label sometimes used to refer to a group that contains exactly one element, more commonly called the trivial group. This one-element structure satisfies the group axioms: the single member serves as the identity element, is its own inverse, and the binary operation returns that element for any pair of inputs. Because of these properties, the trivial group has order 1, is cyclic (generated by its single element), and is abelian.

The trivial group is unique up to isomorphism: any two groups with one element are isomorphic in a canonical way. It plays a special role in algebra and category theory. As an object in the category of groups, it is both initial and terminal (a zero object), meaning there is exactly one group homomorphism from it to any group and exactly one homomorphism from any group to it. Common constructions interact simply with the trivial group: the direct product of any group with the trivial group is isomorphic to the original group, and the quotient of a group by itself yields the trivial group.

Beyond abstract algebra, the term singlegroup can appear informally in other fields to denote a singleton set treated with group-like structure or to describe a collection containing only one subgroup or unit; usage in such contexts is nonstandard and should be clarified by authors.