Read: Continue with Section 3.1.
Turn in: 3.7, 3.12(you can ignore the part about clones), 3.13, 3.14, 3.15, 3.16
- Note that there are a few theorems that I am not asking you to prove, but you are still allowed to use them. In particular, Theorem 3.6 is usually the preferred way to determine if a nonempty subset is a subgroup or not.
- Even if 3.7 seems obvious, please write the proof out carefully (though it will likely be very short).
- In 3.15 and 3.16 please use Theorem 3.6 (or the definition of a subgroup) to prove or disprove if the set is a subgroup.
Extra practice: 3.8, 3.9, 3.10, 3.11
- 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 $S \subseteq H \le G \implies S \subseteq \langle S \rangle \le H \le G$ (remembering that we use $A\le B$ to mean “$A$ is a subgroup of $B$”.)
