Group Theory/Normal subgroups and the Noether isomorphism theorems

In general, the subgroup product is not a subgroup. However, if one of the subgroups involved in the product is a normal subgroup, then it is:

Exercises

 * 1) Prove that the intersection of normal subgroups is again normal.
 * 2) Let $$G$$ be a group, and let $$H_1, \ldots, H_n \trianglelefteq G$$ such that $$[G: H_1], \ldots, [G: H_n]$$ are pairwise coprime. Prove that $$G / (H_1 \cap \cdots \cap H_n) \cong G/H_1 \times \cdots \times G/H_n$$.