$\mathbb{Q} $ is not a projective $\mathbb{Z} $ module

Question

Prove that $\mathbb{Q} $ is not a projective $\mathbb{Z} $ module

Let on the contrary it is projective.

Then $P=\mathbb{Q}$. Then bottom row is given to be exact. Now h is given to be an $\mathbb{Z}$-module homomorphism such that gh=f. But I am not able to think about what contradiction I can obtain.

I am not very good in projective modules.