# Dihedral Group - Internal Direct Product

I have to prove that $D_4$ cannot be the internal direct product of two of its proper subgroups.Please suggest.

Since the order of the group is $8$. Internal direct is possible if there exists two normal subgroups $H$ and $K$ of $D_4$ such that $D_4 = H \times K$.

Then, by Lagranges Theorem we can have $|H| = 2$ and $|K| = 4$ or vice a versa. I can see that both $H$ and $K$ are abelian groups. How to proceed further in this ??

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The following theorem may help: Let $H,K$ be subgroups of $G$. Then if $H \cap K$ is the trivial subgroup, $HK =G$ and $H,K$ are normal in $G$, then $G \cong H \times K$. –  Benja Sep 23 '11 at 11:56
@Tav: Let $(h, k)$ and $(h', k')$ be two elements in $H \times K$. All you need to do is show that $(h, k) \cdot (h', k') = (h', k') \cdot (h, k)$, assuming $H$ and $K$ are abelian groups. –  Zhen Lin Sep 23 '11 at 12:42