Read: Section 5.3 and start 6.2. Also consider watching the following Socratica video: Direct Products of Groups (Abstract Algebra) | Socratica (8:55 min)
Turn in: 5.32, 5.34, 5.36 (just “backwards” direction—see below), 6.2, 6.9
- 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 can either directly argue that $bA = Ab$ for every $b\in B$ or, if you notice that 5.33 applies, just use that. Next 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 5.36, just prove that “if $gHg^{-1} \subseteq H$ for all $g\in G$, then $H \trianglelefteq G$.” And to clarify the hypothesis, $gHg^{-1} \subseteq H$ means that for all $h\in H$, $ghg^{-1} \in H$ and this in turn means that for all $h\in H$ there exists $k \in H$ such that $ghg^{-1} = k$.