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I stuck on the following problem from Hungerford's Algebra.

Let $H,K,N$ be normal in a group $G$ such that $G = H \times K$. Show $N$ is in the center of $G$ or intersects $H$ or $K$ non trivially.

I tried to construct some type of group action on $N$ but it didn't lead anywhere.

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I think that if two normal groups $H$ and $N$ have trivial intersection, then $HN=1$. Can you think of what we might do with that? (I'm not sure about that, but am more confident that $hn=nh$, which might be good enough). –  Eric Stucky Nov 3 '12 at 21:04

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up vote 3 down vote accepted

Suppose that $N\cap H = N\cap K = \{1\}$ and let $hk\in N$. Try showing that $h\in Z(H)$ and $k\in Z(K)$. (For example, if $h\notin Z(H)$, you can derive a contradiction to $N\cap H = \{1\}$.)

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Indeed that works –  Digital Gal Nov 3 '12 at 21:21
    
Is there some deeper theorem (general technique) that can prove this? –  Digital Gal Nov 3 '12 at 21:22

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