Cyclic Group
A group is called a cyclic group if, for some
, every element
is of the form
, where
is some integer. The element
is called
generator of
.
There may be more than one generator of a cyclic group. If is a cyclic group generated by
, then we shall write
or
. The elements of
will be of the form
. Of course, they are not necessarily all distinct.
Examples:
(1) The multiplicative group is cyclic. The generators are
and
.
(2) The multiplicative group is cyclic. We can write
. The generators are
and
.
(3) The multiplicative group of , nth roots of unity are cyclic, a generator being