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- November 1st 2008, 03:58 AM #1

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## need more help ...

is the set of all elements in whose order is a power of . The subgroups of are called the

*primary components*of .

1. Let and be finite abelian groups, and let be a homomorphism. Show that , for each .

2. Let and be finite abelian groups; show that if and only if for all primes .

3. Let be a finite abelian group. Show that any two decompositions of G into direct sums of primary cyclic groups have the same number of summands of each order.

- November 1st 2008, 06:27 PM #2

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- November 1st 2008, 08:17 PM #3

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