is the unique group of group order 5, which is Abelian. Examples include the point group and the integers mod 5 under addition (). No modulo multiplication group is isomorphic to .
The cycle graph is shown above, and the cycle index
The elements satisfy , where 1 is the identity element.
Its multiplication table is illustrated above and enumerated below.
1 | |||||
1 | 1 | ||||
1 | |||||
1 | |||||
1 | |||||
1 |
Since is Abelian, the conjugacy classes are , , , , and . Since 5 is prime, there are no subgroups except the trivial group and the entire group. is therefore a simple group, as are all cyclic graphs of prime order.