Cyclic Group C_5

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C_5 is the unique group of group order 5, which is Abelian. Examples include the point group C_5 and the integers mod 5 under addition (Z_5). No modulo multiplication group is isomorphic to C_5.

The cycle graph is shown above, and the cycle index

The elements A_i satisfy A_i^5=1, where 1 is the identity element.

Its multiplication table is illustrated above and enumerated below.

Since C_5 is Abelian, the conjugacy classes are {1}, {A}, {B}, {C}, and {D}. Since 5 is prime, there are no subgroups except the trivial group and the entire group. C_5 is therefore a simple group, as are all cyclic graphs of prime order.

See also

Cyclic Group, Cyclic Group C2, Cyclic Group C3, Cyclic Group C4, 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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Cite this as:

Weisstein, Eric W. "Cyclic Group C_5." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CyclicGroupC5.html

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