Transitive resolvable idempotent symmetric quasigroups and large sets of Kirkman triple systems Original Research Article

Denniston (J. Combin. Theory Ser. A 26 (1979) 298) presented a tripling construction for large sets of Kirkman triple systems (LKTS) under the condition that there exists a transitive Kirkman triple system (TKTS). In this paper, we first define a transitive resolvable idempotent symmetric quasigroup (TRISQ), and show that a TRISQ of order v always exists for any positive integer v≡3 (mod 6). Then, we present a tripling construction of LKTS by using TRISQ instead of TKTS. Hence the condition “there exists a TKTS” in Dennistonʹs tripling construction can be removed.