Sections 8.1-8.2 Due June 8th

1. I didn't understand how on the bottom of page 245 proving that the elements of Z_6 can be written as a sum of elements of M and N in one and only way proved that they were normal subgroups. The proofs in section 8.2 were very long and complicated, so I wonder whether I will be able to use them sufficiently in homework problems. I'm not understanding how we go about finding the elementary and invariant divisors.
2. It was interesting how prime power orders have such an effect on the groups and subgroups.