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.