Cyclic Group C_2

The group C_2 is the unique group of group order 2. C_2 is both Abelian and cyclic. Examples include the point groups C_s, C_i, and C_2, the integers modulo 2 under addition (Z_2), and the modulo multiplication groups M_3, M_4, and M_6 (which are the only modulo multiplication groups isomorphic to C_2).

The group C_2 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 A_i satisfy A_i^2=1, where 1 is the identity element.


Its multiplication table is illustrated above and enumerated below.


The conjugacy classes are {1} and {A}. The only subgroups of C_2 are the trivial group {1} and entire group {1,A}, both of which are trivially normal.

The irreducible representation for the C_2 group is {1,-1}.

See also

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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The Klein Four. "Finite Simple Group (of Order Two)."

Cite this as:

Weisstein, Eric W. "Cyclic Group C_2." From MathWorld--A Wolfram Web Resource.

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