A finite group is isomorphic to the direct product of two normal subsets with trivial intersection

Question

I have to show the following:

Let $G$ be a finite group and $N_1 , N_2$ normal in $G$. Then $G\cong N_1\times N_2$ if and only if $N_1\cap N_2 = \{e\}$.

I have no idea on either direction, so I would be grateful for any little hint!

Thank you!

Answer 1

This is not true unless $N_1,N_2$ generates $G$, in this case let $p_i:G\rightarrow G/N_i$ the quotient map, show that $p_1$ iduces an isomorphism $N_2\rightarrow G/N_1$. Consider $f:G\rightarrow N_1\times N_2$ defined $f(x)=(p_1(x),p_2(x))$ show that it is an isomorphism.