It is known that for $p\neq 2$, the cyclic group $\mathbb{Z}_p$ does not act freely on $\mathbb{C}P^n$,($n$ odd) and $S^{2m}$. I think then it is true that $\mathbb{Z}_p$ does not act freely on $\mathbb{C}P^n \times S^{2m}$. But I don't know how to prove it.

It does not follow from Euler characteristic. I tried to use Lefschetz fixed point theory but could not solve it. Any solution will be helpful.

Thanks in advance.