In a problem from a past exam I am asked "When can $D_n = \langle r,s\mid r^n = s^2 = (rs)^2 = 1\rangle$, the dihedral group of order $2n$, be expressed as a direct product $G\times H$ of two nontrivial groups?" My answer is when $n=2$.
My reasoning is as follows: what I should seek is nontrivial proper normal subgroups $G, H$ of $D_n$ such that $GH=D_n$ and $G\cap H$ is trivial. A nontrivial proper normal subgroup of $D_n$ is cyclic, if $n\in 1+2\mathbb Z$. If $n \in 2\mathbb Z$, subgroups of the form $\langle r^2,s\rangle$ or $\langle r^2, sr\rangle$ are also normal. However, no matter how we choose $G,H$ from these, we cannot satisfy both $GH=D_n$ and $G\cap H = \{1\}$, if $n>2$.
I would be grateful if you could tell me this is correct.
I think you make too many assertions which you have not established. Note, for example, that if $n=9$, the subgroup generated by $<r^3,s>$ is normal. Also you have to establish "no matter how we choose $G,H$ from these $\dots$"
I think you would do better to notice that in the direct product $G\times H$ every element of $G$ commutes with every element of $H$. This is a property of the direct product which you have not noted in your attempt. $D$ is not commutative, so this condition puts a restriction on what $G$ and $H$ might be, and the elements they might contain.