The cyclic group
is one of the two Abelian groups of group order 9
(the other order-9 Abelian group being ; there are no non-Abelian groups of order 9). An
example is the integers modulo 9 under addition (). No modulo multiplication
group is isomorphic to . Like all cyclic groups, is Abelian.

The numbers of elements satisfying for , 2, ..., 9 are 1, 1, 3, 1, 1, 3, 1, 1, 9.

Because the group is Abelian, each element is in its own conjugacy class. There are three subgroups: ,
and . Because the group is Abelian, these are
all normal. Since
has normal subgroups other than the trivial subgroup and the entire group, it is
not a simple group.