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.
Its multiplication table is illustrated above.
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.