 TOPICS  # Cyclic Group C_2

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 .

Cyclic Group, Cyclic Group C3, Cyclic Group C4, Cyclic Group C5, Cyclic Group C6, Cyclic Group C7, Cyclic Group C8, Cyclic Group C9, Cyclic Group C10, Cyclic Group C11, Cyclic Group C12

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## References

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

## Cite this as:

Weisstein, Eric W. "Cyclic Group C_2." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CyclicGroupC2.html