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."
The cycle graph is shown above, and the cycle index is
 |
The elements 
satisfy 
,
where 1 is the identity element.
Its multiplication table is illustrated above and enumerated below.
 | 1 |  |
| 1 | 1 |  |
 |  | 1 |
The conjugacy classes are 
and 
. The only subgroups of 
are the trivial group 
and entire group 
, both of which are trivially normal.
The irreducible representation for the 
group is 
.
