TOPICS
Search

Cyclic Group C_5


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.

CyclicGroupC5CycleGraph

The cycle graph is shown above, and the cycle index

 Z(C_5)=1/5x_1^5+4/5x_5.

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

CyclicGroupC5Table

Its multiplication table is illustrated above and enumerated below.

C_51ABCD
11ABCD
AABCD1
BBCD1A
CCD1AB
DD1ABC

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

Explore with Wolfram|Alpha

Cite this as:

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

Subject classifications