Directions.
- Get the template for this assignment (or use your own). Here’s how to do it:
- Use $\LaTeX$ to type up your proofs of the Propositions to Prove listed below. Make sure to use complete sentences and appropriate punctuation. Also, make sure to edit for typos.
- Click on the “Download PDF” button (to the right of the “Recompile” button). Save the file somewhere you can easily find it. Submit the pdf in Canvas.
- Let me know if you have any questions!
Remember, if you have trouble finding the command for a math symbol you want to use, try googling it, email me, or try looking in this Short Math Guide for $\LaTeX{}$.
Propositions to Prove.
- Consider $(\mathbb{R}^3,+)$, where $\mathbb{R}^3$ is the set of all 3-entry row vectors with real number entries and + is ordinary vector addition. Let $K$ be the subset of $\mathbb{R}^3$ consisting of vectors whose entries sum to 0; that is $K:=\{(a,b,c)\in \mathbb{R}^3\mid a+b+c = 0\}$. Prove that $K$ is a subgroup of $(\mathbb{R}^3,+)$.
- Notes. See Problem 3.16(b). 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 $(-a,-b,-c)$.
- For any $n\in \mathbb{Z}$, define $n\mathbb{Z} := \{nk\mid k\in \mathbb{Z}\}$. Prove that $n\mathbb{Z}$ is a subgroup of $(\mathbb{Z},+)$.
- Notes. See Problem 3.17(c). You do not need to prove that $(\mathbb{Z},+)$ is a group. You can use, without proof, that $0$ is the identity of $(\mathbb{Z},+)$ and that the inverse of $a$ is $-a$.
- Prove that if $G$ is a group, then $Z(G)$ is a subgroup of $G$.
- Notes. See Theorem 3.23. I’m only asking you to prove that $Z(G)$ is a subgroup; you don’t need to prove that it’s abelian like was asked in Theorem 3.23.
