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