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This question already has an answer here:

Hi I'm having trouble understanding why the quaternionic group cannot be written as a semi-direct product in a non-trivial way. Is there a proof of why this is the case?

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marked as duplicate by Dietrich Burde, Najib Idrissi, Namaste group-theory Sep 22 '14 at 12:33

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The only non trivial subgroups of $\mathbf{Q}$ are $\{1,-1\}$, $\{i\}$, $\{j\}$ and $\{k\}$. Since one needs at least one group of order $2$ in the decomposition $\mathbf{Q}=GH$ if must be $\{1,-1\}$, but this group is the center of $\mathbf{Q}$ so can not be a factor of the decomposition.

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