Finite projective planes and symmetric divisible designs

证明了任意对称横向设计STD_2的自同构群的阶性[12]；对于一些非负整数α和β， 6]等于2^α3^β。证明了如果π是一个允许9阶共线群G的12阶投影平面，则G是一个初等阿贝尔群，π不是平面。我们将STD_6[18；3] s和STD_7[21；[3]在点和块上都有一个阶为9的半正则非同构自同构群。我们证明了非同构STD_8[24；[3]至少是24。

We proved that the order of an automorphism group of any symmetric transversal design STD_2[12 ; 6] is 2^α3^β for some nonnegative integers α and β. We proved that ifπis a projective plane of order 12 admitting a collineation group G of order 9, then G is an elementary abelian group andπis not planar. We classified STD_6[18 ; 3]'s and STD_7[21 ; 3]'s that have a semiregular nonisomorphic automorphism group of order 9 on both points and blocks containing an elation of order 3. We showed that the number of nonisomorphic STD_8[24 ; 3]'s is at least 24.