Collineations and generalized incidence matrices

D. R. Hughes
1957 Transactions of the American Mathematical Society  
Introduction. In an earlier paper [12] the author has found certain numerical conditions which must be satisfied if a (v, k, X) configuration is to possess a regular collineation group. These conditions were derived from relations similar to the incidence matrix equations of [3; 4]. Here these conditions are extended in that we show the existence of certain matrix equations which must be satisfied if a iv, k, X) configuration is to possess an arbitrary collineation group. Using these equations,
more » ... ng these equations, it is shown that the number of transitive classes of points is always equal to the number of transitive classes of lines (2) , and that the matrix equations always lead to a rational congruence. Specializing the collineation groups considered to the class of standard collineation groups, we apply the Hasse-Minkowski theory to the rational congruence mentioned above and derive numerical conditions on the parameters v, k, X, the order m of the collineation group, and the number N of fixed points. A standard collineation group is one whose nonidentity elements all fix the same set of points and lines; any collineation group of prime order is standard, so a (v, k, X) configuration with nontrivial collineations possesses nontrivial standard collineation groups. The author wishes to take this opportunity to thank H. J. Ryser for his helpful comments in the preparation of this part of the paper, in particular with the application of the Hasse-Minkowski theory. Several interesting but unsolved problems in the theory of groups arise from these investigations. An intimate connection is displayed between the theories of collineations of (v, k, X) configurations, groups possessing subsets similar to partial difference sets, and topics in the theory of matrices with integer elements. 2. Collineation groups. Let v, k, X, where v>k>\>0, be integers satisfying \(v -1) =k(k -l). Let rr be a collection of v points and v lines, together with an incidence relation (i.e., point on line, line contains point, etc.) such that each point (line) is on k lines (contains k points) and such that each pair of distinct points (lines) are on exactly X common lines (contain exactly X
doi:10.1090/s0002-9947-1957-0093730-4 fatcat:fvep32yjbfhllbw7sxz2wwtoim