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.
