Read: Continue with Section 3.1.
Turn in: 3.16, 3.17(a,b,c), 3.18, 3.22
- On 3.16 and 3.17, please use Theorem 3.6 to prove or disprove if the set is a subgroup.
- Also, for 3.16, you do not need to prove that \((\mathbb{R}^3,+)\) is a group. You can use, without proof, that \((0,0,0)\) is the identity of \((\mathbb{R}^3,+)\) and that the inverse of \((a,b,c)\) is \((-a,-b,-c)\).
- On 3.17, you can use facts we already know about $(\mathbb{Z},+)$: $0$ is the identity and the inverse of $a$ is $-a$.
Extra practice: 3.17(d,e), 3.19, 3.20, 3.21
- Your proof of 3.21 will likely be short. You are given that $H$ is a subgroup of $G$ and that $G$ is abelian (i.e. the operation is commutative); you then need to prove that $H$ is abelian. The key point is that when we say $H$ is a subgroup of $G$ we are using the same operation for both $H$ and $G$.