The cyclic group is the unique Abelian group of group order 10 (the other order-10 group being the non-Abelian ). Examples include the integers modulo 10 under addition () and the modulo multiplication groups and (with no others). Like all cyclic groups, is Abelian.

The cycle graph of is shown above. The cycle index is

Its multiplication table is illustrated above.

The numbers of elements satisfying for , 2, ..., 10 are 1, 2, 1, 2, 5, 2, 1, 2, 1, 10.

Because the group is Abelian, each element is in its own conjugacy class. There are four 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.