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

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.