The group
is the unique group of group order 2. is both Abelian and cyclic. Examples include the point
groups ,
, and , the integers modulo 2 under addition (), and the modulo
multiplication groups , , and (which are the only modulo multiplication groups isomorphic
to ).

The group
is also trivially simple, and forms the subject for
the humorous a capella song "Finite Simple Group (of Order 2)" by the Northwestern
University mathematics department a capella group "The Klein Four."