A cyclic group is a group generated by a single element. All the elements of the group are formed by repeated composition of some element
with itself, including the identity element.
If the group
has order
we may write
This necessarily means that all elements of cyclic groups commute and and that cyclic groups abelian, since if
and
for some
so that
This then means that the Cayley table has a line of symmetry about the leading diagonal, as shown below for the rotation group of a regular hexagon.
The abelian property is inherited by all subgroups of
as is the symmetry property of the Cayley table.