Is it true that for $p\neq 2 $,   $\mathbb{Z}_p$ does not act freely on  $ \mathbb{C}P^{n_1} \times \mathbb{C}P^{n_2}  \times  \cdots  \times  \mathbb{C}P^{n_k}$? 

How to prove it? 

Thank you so much in advance.