is the unique group of group
order 3. It is both Abelian and cyclic.
Examples include the point groups 
, 
, and 
and the integers under addition modulo 3 (
). No modulo multiplication
groups are isomorphic to 
.
The cycle graph of 
is shown above, and the cycle
index is
 |
The elements 
of the group satisfy 
where 1 is the identity element.
Its multiplication table is illustrated above and enumerated below (Cotton 1990, p. 10).
 | 1 |  |  |
| 1 | 1 |  |  |
 |  |  | 1 |
 |  | 1 |  |
Since 
is Abelian, the conjugacy classes are 
, 
, and 
. The only subgroups of 
are the trivial group 
and the entire group, which are both
trivially normal. 
is therefore a simple group, as are all cyclic graphs
of prime order.
The irreducible representation (character table) is therefore
 | 1 |  |  |
 | 1 | 1 | 1 |
 | 1 | 1 |  |
 | 1 |  | 1 |
