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Let $N_1$ and $N_2$ be normal subgroups of $G$.
Prove that $N_1N_2/(N_1\cap N_2) \cong (N_1N_2/N_1)\oplus (N_1N_2/N_2)$.
I think the homomorphism must be $\phi : N_1N_2 \to (N_1N_2/N_1)\oplus (N_1N_2/N_2)$
such that $\phi(n_1n_2)=(n_1n_2N_1,n_1n_2N_2)$
Then by 1st isomorphism theorem, $N_1N_2/ker\phi \cong Im\phi$
and $ker\phi = N_1 \cap N_2$
But it is hard to show that $\phi$ is surjective.
What should I do?