## Commutativity of Cyclic Groups

A cyclic group is a group generated by a single element. All the elements of the group are formed by repeated composition of some elementwith itself, including the identity element.

If the grouphas orderwe may write

This necessarily means that all elements of cyclic groups commute and and that cyclic groups abelian, since ifandfor someso 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 ofas is the symmetry property of the Cayley table.