Abstract Algebra: Theory and Applications

State Lagrange’s Theorem in your own words.

Determine the left cosets of $⟨3⟩$ in ${\mathbb{Z}}_9\text{.}$

The set $\{(),(12)(34),(13)(24),(14)(23)\}$ is a subgroup of $S_4\text{.}$ What is its index in $S_4\text{?}$

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

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