Read: Sections 5.2 and 5.3. Also consider watching the following Socratica video:
Turn in: 5.17, 5.19, 5.20, 5.21, 5.24, 5.32, 5.34, 5.36 (just “backwards” direction—see below)
- Please try 5.16, even though I’m not asking you to turn it in. We’ll work through it in class together. Consider using 5.11 and 5.13.
- For 5.17, use 5.16.
- For 5.19, think back to the definition of the order of an element, which is about the order of a subgroup.
- On 5.21, see if you can first use Lagrange’s theorem to prove that Theorem 4.10 applies to $G$. Once you know $G$ is cyclic, then you can use Theorem 4.40 to finish up.
- Before starting 5.24, read Definition 5.23 a couple of times. The comment that $[G:H] = |G|/|H|$ when $G$ is a finite group is very useful.
- Here’s a hint for 5.34. Let’s name the subgroups: $A = \langle s \rangle$, $B = \langle r^2, sr^2\rangle$. To show $A\trianglelefteq B$, you need to show $bA = Ab$ for every $b\in B$; however, if you see that 5.33 applies (wink, wink), you can use that. Next, you should explain why $B \trianglelefteq D_4$. Finally, to show $B\not\trianglelefteq D_4$, you only need to find one element $g\in D_4$ such that $gB \neq Bg$.
- For 5.26, just prove that “if $gHg^{-1} \subseteq H$ for all $g\in G$, then $H \trianglelefteq G$.”
Extra practice: 5.25, 5.30, 5.31, 5.33, 5.38
