Subgroup of a nilpotent group Let $G$ be nilpotent and $H \le G$. Let $P_1,P_2,\ldots,P_k$ be the Sylow subgroups of $H$. Is it true that $H = P_1 P_2 \cdots P_k$?
I know that when $G$ is nilpotent, it is the direct product of its Sylow subgroups, but is that true for a subgroup of $G$ as well?

Yes,because a subgroup of a nilpotent group is also nilpotent.
Let $1=G_0\leq G_1\leq ...\leq G_n=G$  be a central series of $G$ that means $[G_i,G]\leq G_{i-1}$.
Let $H_i=H\cap G_i$ .Then $1=H_0\leq H_1\leq....\leq H_n=H$  is a central series of $H$, because $[H_i,H]=[H\cap G_i,H]\leq H\cap [G_i,G]\leq H\cap G_{i-1}=H_{i-1}$
