is one of the two groups of group
order 4. Like ,
it is Abelian, but unlike , it is a cyclic.
Examples include the point groups (note that the same notation is used for the abstract cyclic
group
and the point group isomorphic to it) and , the integers modulo 4 under addition (), and the modulo
multiplication groups and (which are the only two modulo multiplication groups
isomorphic to it).

The multiplication table for this group may be written in three equivalent ways by permuting the symbols used for the group elements
(Cotton 1990, p. 11). One such table is illustrated above and enumerated below.

1

1

1

1

1

1

The conjugacy classes of are , , , and . In addition to the trivial group and the entire group,
also has as a subgroup which, because the group is Abelian, is
normal.
is therefore not a simple group.

Elements
of the group satisfy ,
where 1 is the identity element, and two of the
elements satisfy .

The group may be given a reducible representation using complex
numbers