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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 $G \cong H \times K$.2011-09-23

1 Answers 1

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A hint: the direct product of abelian groups is abelian.

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    @JackSchmidt: Thanks for your help. I appreciate it.2011-09-23