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 $H_1,\dots ,H_n⊴G$ such that $[G:H_1],\dots ,[G:H_n]$ are pairwise coprime. Prove that $G/(H_1\cap \cdots \cap H_n)\cong G/H_1\times \cdots \times G/H_n$.