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Cyclic Group C_3


C_3 is the unique group of group order 3. It is both Abelian and cyclic. Examples include the point groups C_3, C_(3v), and C_(3h) and the integers under addition modulo 3 (Z_3). No modulo multiplication groups are isomorphic to C_3.

CyclicGroupC3CycleGraph

The cycle graph of C_3 is shown above, and the cycle index is

 Z(C_3)=1/3x_1^3+2/3x_3.

The elements A_i of the group satisfy A_i^3=1 where 1 is the identity element.

CyclicGroupC3Table

Its multiplication table is illustrated above and enumerated below (Cotton 1990, p. 10).

C_31AB
11AB
AAB1
BB1A

Since C_3 is Abelian, the conjugacy classes are {1}, {A}, and {B}. The only subgroups of C_3 are the trivial group {1} and the entire group, which are both trivially normal. C_3 is therefore a simple group, as are all cyclic graphs of prime order.

The irreducible representation (character table) is therefore

Gamma1AB
Gamma_1111
Gamma_211-1
Gamma_31-11

See also

Cyclic Group, Cyclic Group C2, 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

Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, 1990.

Cite this as:

Weisstein, Eric W. "Cyclic Group C_3." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CyclicGroupC3.html

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