### 1.

State Sylow's First Theorem.

### 2.

How many groups are there of order $$69\text{?}$$ Why?

### 3.

Give two descriptions, fundamentally different in character, of the normalizer of a subgroup.

### 4.

Suppose that $$G$$ is an abelian group. What is the commutator subgroup of $$G\text{,}$$ and how do you know?

### 5.

What's all the fuss about Sylow's Theorems?