Biplanes (56,11,2) with automorphisms of order 4 fixing some point (Q1113907)

The authors show that all biplanes of order 9 (i.e., all symmetric designs \(S_ 2(2,11,56))\) which admit an automorphism of order 4 fixing one point are known. One obtains the 4 examples which have been known for some time [cf. e.g. \textit{C. J. Salwach} and \textit{J. A. Mezzaroba}: ``The four known biplanes with \(k=11\), Intern. J. Math. Math. Sci. 2, 251-260 (1979; Zbl 0411.51006)]. (There is a fifth biplane recently obtained by \textit{Z. Janko} and \textit{T. van Trung} [``A new biplane of order 9 with a small automorphism group'', J. Comb. Theory, Ser. A 42, 305-309 (1986)] but it does not admit such an automorphism of order 4.) The proof uses standard arguments and a computer.