The Structure of Certain Triple Systems

Abstract

For each prime power $q \equiv 7 \pmod {12}$, there is a triple system of order q whose automorphism group is transitive on unordered pairs. The object of this paper is to study these systems. This is done by analyzing how pairs of elements are linked. The linkage of a and b consists of a triple (a, b, c) and of some cycles in which adjacent pairs of elements form triples alternately with a and with b. Because of the transitivity, the lengths of the cycles will be independent of the choice of a and b. Using a computer, the linkage between two elements was determined for each $q < 1000$ . Some curious facts concerning the lengths of the cycles were uncovered; for example, the number of cycles of length greater than 4 is even. The systems of prime order $p < 1000$ were found to have no proper subsystems of order greater than 3. In the remaining case, $q = 343$, there are subsystems of orders 7 and 49, and all subsystems of the same order are isomorphic. For no q with $7 < q < 1000$ is the automorphism group doubly transitive. Finally, some general results are proved. The cycles of lengths 4 and 6 are determined. Using this result, it is shown that there can be no subsystem of order 7 or 9, except for the subsystems of order 7 when q is a power of 7. Hence, by a theorem of Marshall Hall, the automorphism group cannot be doubly transitive, except possibly when q is a power of 7. (Added August 1974. In a postscript, it is shown that the automorphism group is not doubly transitive in this case either.)