Cyclic Group Is Abelian Integers

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

$cyclic group is abelian integers$

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

$\fn_cm G =< a:a^{2n}=e>$

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.

$\fn_cm e = a^{12},a^{1},a^{2},a^{3},a^{4}........,a^{11}$

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.

1,2,3,4,5………..12

Therefore, G has

The subgroup of order 1 Which is {e}

The subgroup of order 2 Which is $\fn_cm \left \{ e,a^{6} \right \}$

The subgroup of order 3 Which is $\fn_cm \left \{ e,a^{4},a^{8} \right \}$

The subgroup of order 4 which is $\fn_cm \left \{ e,a^{3}a^{6},a^{9} \right \}$

The subgroup of order 6 which is $\fn_cm \left \{ e,a^{2},a^{4},a^{6},a^{10} \right \}$

The subgroup of order 12 Which is G itself.

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