How to prove a subgroup is normal?

Question

Prove that $D$ is a normal subgroup of $C$ if $C=S_3 \times \Bbb Z_4$ and $D=\langle((132),2)\rangle$. I know to prove a subgroup is normal you have to show aH=Ha, but I'm just not sure how to do this particular example. My thought was to to show that (132) is normal in S3 and 2 is normal in Z4 because multiplication commutes, but I picked (12) to multiply with (132), and if my math is correct (12)(132) is not equal to (132)(12). Anyway would be very helpful to see how to do this properly. Thanks in advance.

Answer 1

You're almost there. What is the index of $H = \langle (132) \rangle$ in $S_3$? What does this tell you about $H$?

Also, your reasoning is slightly flawed in trying to show that $H$ is normal. If $H$ is normal in $S_3$, it is not necessarily the case that $(12)(132) = (132)(12)$. Instead, it would be the case that $(12)(123) = h(12)$ for some $h \in H$.

In general, $H$ is normal if and only if, for any $g \in S_3$ and $h \in H$, you have $ghg^{-1} \in H$. This is equivalent to showing $gH = Hg$ for all $g \in G$.