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 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.