Let G be a cyclic group is Abelian integers, generated by a (constant element) and x, y (variables) ∈ G. Then there are positive integers k and m such that

so

Thus, the group g is an Abelian group.

EXAMPLE:

If G is a cyclic group of even order, then prove that there is an only one subgroup of order 2 in G.

SOLUTION:

Let

Be a cyclic group of order 2n, where n is a positive integer.

By proof, result G be a cyclic group of order n generated by a.then, for each

positive divider d of n, there is a unique subgroup (of G) of order d.

If a positive integer d divide |G|, then G has exactly one subgroup of order d.

Now

|G| = 2n and 2 divides 2n, so G has only one subgroup of order 2.

EXAMPLE:

Find all the subgroup of a cyclic group of order 12.cyclic group is abelian integers

Solution:

Let G be a cyclic group of order 12 and ‘a’ be a generator. Then the elements of G are.

By theorem

G be a cyclic group of order n generated by a.then, for each

positive divider d of n, there is a unique subgroup (of G) of order d.