### 1.

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

### 2.

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

### 3.

Find a subgroup of order 6 in $$\mathbb Z_8\times\mathbb Z_3\times\mathbb Z_3\text{.}$$

### 4.

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?

### 5.

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