# Description of Abelian group

Modern Algebra

Group Theory (VII)

To prove that if G is an abelian group, then for all a,b belongs to G and all integers n, (a.b)^n=a^n.b^n.

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Prove that if is an abelian group, then for all and all integers , .

Proof:- Let be arbitrary.

By hypothesis, is an abelian group,

which imply that ---------------------------------(1)

which imply that

and

For

Suppose where is any integer, then

For

Hence by mathematical induction,

for all ...

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This shows how to complete a proof regarding an Abelian group. The solution is detailed and well presented.