Properties of Cyclic Groups
Theorem 1: Every cyclic group is abelian.
Proof: Let be a generator of a cyclic group
and let
for any
then


Thus the operation is commutative and hence the cyclic group is abelian.
Note: For the addition composition the above proof could have been written as (addition of integer is commutative)
Thus the operation + is commutative in .
Theorem 2: The order of a cyclic group is the same as the order of its generator.
Proof: Let the order of a generator of a cyclic group be
, then


When (say)
We observe that
Thus there are exactly elements in the group by
, where
. Therefore there are
and only
distinct elements in the cyclic group, i.e. the order of the group is
.
Theorem 3: The generators of a cyclic group of order are all the elements
,
being prime to
and
.
Proof:
We know that
Therefore the order of is
.
Also if
, because
does not divide
, nor does it divide
, therefore it does not divide
.
Now let,
and
Thus, is a generator of the group.