If G is a finite abelian group G| = P1 ki . P's mks is the decomposition of G| into a product of distinct prime numbers, then G ~ G1 X .

By using induction and results from Q5&Q6 

Prove the following result:

If G is a finite abelian group |G| = p1k1 · · · psks is the decomposition of |G| into a

product of distinct prime numbers, then G ≃ G1 × · · · × Gs where Gj is an abelian

group of order pjkj.

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If G is a finite abelian group G| = P1ki . . . P'smks is the decomposition of G| into aproduct of distinct prime numbers, then G ~ G1 X . . . XGs where G; is an abeliangroup of order Pikj