As you know Lefschetz number is $\tau (f)\doteq \sum_n (-1)^n {\rm tr}(f_\ast : H_n(X)\rightarrow H_n(X))$ where $X$ is a finite simplicial complex and $f : X\rightarrow X$. So LFPT is that if $\tau(f)\neq \emptyset$, then $f$ has a fixed point.
Here I want to know whether or not LFPT holds if the coefficient ring $G$ is ${\bf Z}_p$. Thank in advance.