On the fixed points of an automorphism of a symmetric design (Q1842152)

Using elementary counting arguments, it is proved that if a non-identity automorphism \(\alpha\) of a symmetric \((v,k, \lambda)\) design has \(f\) fixed points, then \(f \leq {1 \over 4} (v + 3k - 6)\) if \(|\alpha |\geq 3\) and \(f \leq {1 \over 3} (v + 2k - 4)\) if \(|\alpha |= 2\) where \(|\alpha |\) denotes the order of \(\alpha\). As a corollary, the following result of Feit is obtained: Let \(\alpha\) be an automorphism of a nontrivial symmetric \((v,k, \lambda)\) design with \(|\alpha |\geq 3\), then \(f \leq{1 \over 2}v\).