6.21 (1978)

G. Higman proved that, for any prime number $p$, there exists a natural number $\chi(p)$ such that the nilpotency class of any finite group $G$ having an automorphism of order $p$ without non-trivial fixed points does not exceed $\chi(p)$. At the same time he showed that $\chi(p) \geqslant (p^2 - 1)/4$ for any such Higman’s function $\chi$. Find the best possible Higman’s function. Is it the function defined by equalities $\chi(p) = (p^2 - 1)/4$ for $p > 2$ and $\chi(2) = 1$? This is known to be true for $p \leqslant 7$.