Read: Start of Section 3.1. Also watch this Socratica video, but please stop watching at 3:52 (if not a little sooner), so Liliana Castro doesn’t spoil anything for you: Cyclic Groups - Abstract Algebra | Socratica (5:00 min)
Turn in: 3.7, 3.12(ignore the parts about clones), 3.14(ignore the part about minimal generating sets), 3.15
- Note that there are a few theorems that I am not asking you to prove, but you are still allowed to use them.
- Theorem 3.6 is often the preferred way to determine if a nonempty subset is a subgroup or not.
- Theorem 3.10 is important, especially the part that says “\(\langle S \rangle\) is the smallest subgroup of \(G\) containing \(S\).” This means that any subgroup \(H\) of \(G\) that contains \(S\) must also contain \(\langle S \rangle\). Or, more symbolically, it means that \(S \subseteq H \le G \implies \langle S \rangle \le H \le G\) (remembering that we use \(A\le B\) to mean “\(A\) is a subgroup of \(B\)”.)
- Even if 3.7 seems obvious, please write the proof out carefully (though it will likely be very short).
- For 3.12, the relevant Cayley diagram for \(\operatorname{Spin}_{1\times 2}\) is in Figure 2.3 on page 34.
- In 3.14–3.15, please use Theorem 3.6 (or the definition of a subgroup) to prove or disprove if the set is a subgroup.
Extra practice: 3.6, 3.9, 3.10, 3.13