Let $G$ be a group with $H_1,\ldots,H_n$ normal subgroups. Define $\varphi:G\mapsto \prod_i G/H_i$ by $\varphi(x)=(xH_1,\ldots,xH_n).$ Prove:
- $\ker(\varphi)=\cap_i^n H_i$,
- If every $H_i$ has finite index in $G$, and $|G/H_i|$ and $|G/H_j|$ are relatively prime for $i\neq j$ then $\varphi$ is a surjection and $$[G:\cap_i^n H_i]=\prod_i |G/H_i|.$$
How do I prove $\varphi$ is a surjection?
Part 1 is quite easy and the last equality follows from applying the first isomorphism theorem to $\varphi$ but I can't figure out how to use the relatively prime hypothesis to prove surjectivity.
Any help would be appreciated.