Herstein problem 2.13.10
Let $G$ be a group, $K_1,..,K_n$ normal subgroups of $G$. Suppose that $K_1\cap K_2\cap…\cap K_n=(e)$. Let $V_i=G/K_i$. Prove that there is an isomorphism of $G$ into $V_1\times V_2\times .. \times V_n$
I tried to prove $G$ is isomorphic to internal direct product of $V_i$'s but this will give an isomorphism onto $V_1\times V_2\times .. \times V_n$ while we are asked to prove merely into.
I am unable to deal with two concepts together: external direct product and quotient group. Please give a hint. Please do not give the solution.
Best Answer
Hint: The embedding: $$g\mapsto (gK_1, gK_2,...).$$