$\left(H_{i}\right)_{i \in I}$ is a family of normal subgroups, Show $G$ is isomorphic to at least one subgroup of $\prod_{i \in I} G / H_{i}$

Question

$\left(H_{i}\right)_{i \in I}$ is a family of normal subgroups of a group $G$ which: $\bigcap_{i \in I} H_{i}=\{e\}$. I have two questions:

1. Show $G$ is isomorphic to at least one subgroup of $\prod_{i \in I} G / H_{i}$
2. Can we consider a situation in which $G$ is isomorphic to $\prod_{i \in I} G / H_{i}$ itself? if YES, what is the situation or the condition? if NO, why?

I've tried for few hours and I've got nothing :) Any help would be appreciated.

Answer 1

HINT

Start with $f:G\to G/H_1\times ...\times G/H_k\times ...$ with $f(g)=(gH_1,...,gH_k,...)$ which has kernel $\cap_iH_i=\{e\}$ hence it is monomorphism.