The cyclic group 
is one of the three Abelian groups of the five groups total of group
order 8. Examples include the integers modulo 8 under addition (
) and the residue classes modulo 17 which have quadratic
residues, i.e., 
under multiplication modulo 17. No modulo
multiplication group is isomorphic to 
.
The cycle graph of 
is shown above. The cycle index
is
 |
Its multiplication table is illustrated above.
The elements 
satisfy 
,
four of them satisfy 
,
and two satisfy 
.
Because the group is Abelian, each element is in its own conjugacy class. There are four subgroups: 
,
, 
, and 
which, because the group is Abelian, are all
normal. Since 
has normal subgroups other than the trivial subgroup and the entire group, it is
not a simple group.
