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.
