Read: Read Section 6.1. Also consider watching the following two Socratica videos:
Turn in: 6.2, 6.9, 6.10, 6.13, 6.15 (just “backwards” direction—see below), 6.18, 6.20
- For 6.15, just prove that “if $m$ and $n$ are relatively prime, then $\mathbb{Z}_{m} \times \mathbb{Z}_{n}$ is cyclic.” Try to explicitly find a generator $(a,b)$ for $\mathbb{Z}_{m} \times \mathbb{Z}_{n}$ using 6.12; a key point is that $m$ and $n$ are relatively prime if and only if $\operatorname{lcm}(m,n) = mn$.
- On 6.18, for each group that is cyclic, please also write down what familiar group it is isomorphic to using 4.40.
Extra practice: 6.11, 6.14, 6.18
