### 1.

State Lagrange's Theorem in your own words.

### 2.

Determine the left cosets of $$\langle 3 \rangle$$ in $$\mathbb Z_9\text{.}$$

### 3.

The set $$\{(), (1\,2)(3\,4), (1\,3)(2\,4), (1\,4)(2\,3)\}$$ is a subgroup of $$S_4\text{.}$$ What is its index in $$S_4\text{?}$$

### 4.

Suppose $$G$$ is a group of order 29. Describe $$G\text{.}$$

### 5.

The number $$p=137909$$ is prime. Explain how to compute $$57^{137909}\pmod{137909}$$ without a calculator.