Steiner Triple Systems of Order 19 with Nontrivial Automorphism Group

There are 172,248 Steiner triple systems of order 19 having a nontrivial automorphism group. Computational methods suitable for generating these designs are developed. The use of tactical configurations in conjunction with orderly algorithms underlies practical techniques for the generation of the designs, and the subexponential time isomorphism technique for triple systems is improved in practice to test isomorphisms of the designs. The automorphism group of each of the triple systems is computed, and a summary presented of the number of systems with each possible type of automorphism. 1. Steiner triple systems and their groups A Steiner triple system of order v , briefly STS(v), is a pair ( V, 38), where F is a set of v elements and 38 is a set of 3-element subsets of V , with the property that every 2-subset of V appears in exactly one subset of 38. Sets in 38 are triples. An automorphism of an STS(v) is a permutation on V that maps each triple in 38 to a triple of 38 , and the automorphism group is the group of all automorphisms of the STS. Steiner triple systems with nontrivial automorphisms have been studied for many reasons. Not least among these is the fact that Steiner triple systems are too numerous to examine exhaustively, even for order 19. In fact, Stinson and Ferch [20] have shown that the number of nonisomorphic STS(19) is at least 2,395,687 and estimate the exact number to be on the order of 109. For this reason, much effort has been concentrated on STS(19) with additional properties. Bays [ 1 ] enumerated the four STS( 19) having a cyclic automorphism in 1932. More recently, Denniston [8] generated the 184 nonisomorphic STS(19) with a reversal, i.e., an automorphism fixing one element and mapping the rest in nine 2-cycles. Phelps and Rosa [13] generated the ten nonisomorphic STS(19) having a 2-rotational automorphism, i.e., an automorphism fixing one element and mapping the remainder in two 9-cycles. Using Gelling's list of 1-factorizations of order ten [9] and properties of their automorphism groups, Stinson and Seah [21] determined that the number of STS(19) having a subsystem of order 9 is precisely 284,457 . See Prince [14, 15] for some related results on the enumeration of STS(19). Received by the editor February 7, 1991. 1991 Mathematics Subject Classification. Primary 05B07. © 1992 American Mathematical Society 0025-5718/92 $1.00+ $.25 per page