Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: On STD6[18, 3]’s and STD7[21, 3]’s admitting a semiregular automorphism group of order 9. | On STD6 18 3 s and STD7 21 3 s admitting a semiregular automorphism group of order 9 Kenzi Akiyama Department of Applied Mathematics Fukuoka University Fukuoka 814-0180 Japan akiyama@ Masayuki Ogawa Computer Engineering Inc. Hikino Yahatanisi-ku Kitakyushu-city Fukuoka 806-0067 Japan a_meteoric_stream_0521@ Chihiro Suetake Department of Mathematics Faculty of Engineering Oita University Oita 870-1192 Japan suetake@ Submitted Sep 11 2009 Accepted Nov 30 2009 Published Dec 8 2009 Mthematics Subject Classifications 05B05 05B25 Abstract In this paper we characterize symmetric transversal designs STD fc u s which have a semiregular automorphism group G on both points and blocks containing an elation group of order u using the group ring Z G . Let n be the number of nonisomorphic STDa 3A 3 s. It is known that n1 1 n2 1 n3 4 n4 1 and n5 0. We classify STD6 18 3 s and STD7 21 3 s which have a semiregular noncyclic automorphism group of order 9 on both points and blocks containing an elation of order 3 using this characterization. The former case yields exactly twenty nonisomorphic STD6 18 3 s and the latter case yields exactly three nonisomorphic STD7 21 3 s. These yield n6 20 and n7 5 because B. Brock and A. Murray constructed two other STD7 21 3 s in 1991. We used a computer for our research. This research was partially supported by Grant-in-Aid for Scientific Research No. 21540139 Ministry of Education Culture Sports Science and Technology Japan. THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R148 1 1 Introduction A symmetric transversal design STDA k u STD is an incidence structure D P B I satisfying the following three conditions where k 2 u 2 and A 1 i Each block contains exactly k points. ii The point set P is partitioned into k point sets P0 P1 Pk-1 of equal size u such that any two distinct points are incident with exactly A blocks or no block according as they are contained in different Pi s or not. Po Pl Pk-1 are .
