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 .

The Klein Four. "Finite Simple Group (of Order Two)." http://www.math.northwestern.edu/~matt/kleinfour/.

CITE THIS AS:

Weisstein, Eric W. "Cyclic Group C2." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CyclicGroupC2.html