Cyclic Group C_4

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 cycle graph of is shown above, and the cycle index is given by

(1)

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

			(2)
			(3)
			(4)
			(5)

or real matrices

			(6)
			(7)
			(8)
			(9)