Challenge problem on my Abstract Algebra Homework that I was unable to make any significant progress on. Any hints for parts (a) and (b) would be much appreciated. It seems that one direction for part (b) is rather trivial, as if $H x K -> G$ is a group isomorphism, then all of those properties should directly follow, but I don't really know how to formally show this. I usually make more progress than this on challenge problems, but this problem has me completely stumped.
part (a) Suppose $H$ is a normal subgroup of $G$. It suffices to show that $HK=KH$ (why?). Let $hk \in HK$. Since $H$ is normal
$$(kk^{-1})hk=k(k^{-1}hk)=kh'$$
for some $h' \in H$ Hence $Hk \subseteq KH$. It follows similarly for $KH \subseteq HK$ and so we have equality. Hence $HK$ is a subgroup of $G$. Let me know if you need help with $HK$ being the join of $H$ and $K$.
part (b) I'll go ahead and assume you can show this is a homomorphism. So, what does it mean for $x$ to be the kernel of this map? $(h,k) \mapsto e$. So, if we assume $H \cap K = \{e\}$, then $hk=e$ \implies $h=k^{-1}$. We conclude $h=k^{-1}=e$ (why?) $G$ being the join of $H$ and $K$ automatically gives surjectivity (why?). Thus, we have an isomorphism.