Abstract Algebra: Theory and Applications

How many abelian groups are there of order $200=2^3{5}^2\text{?}$

How many abelian groups are there of order $729=3^6\text{?}$

Find a subgroup of order 6 in ${\mathbb{Z}}_8\times {\mathbb{Z}}_3\times {\mathbb{Z}}_3\text{.}$

It can be shown that an abelian group of order $72$ contains a subgroup of order $8\text{.}$ What are the possibilities for this subgroup?

What is a principal series of the group $G\text{?}$ Your answer should not use new terms defined in this chapter.