Combinatorial Designs—A Tribute to Haim Hanani by A. Hartman

By A. Hartman

Haim Hanani pioneered the options for developing designs and the speculation of pairwise balanced designs, prime on to Wilson's life Theorem. He additionally led the way in which within the examine of resolvable designs, masking and packing difficulties, latin squares, 3-designs and different combinatorial configurations. The Hanani quantity is a suite of analysis and survey papers on the vanguard of study in combinatorial layout idea, together with Professor Hanani's personal most up-to-date paintings on Balanced Incomplete Block Designs. different components coated comprise Steiner platforms, finite geometries, quasigroups, and t-designs.

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Yx = x ) is n = 0 or 1 (mod 4). Several authors investigated J(xy yx = x ) including D . A . Norton and S . K . K. D. C. Lindner, N . S . Mendelsohn and S . R . Sun [47]. The most conclusive result was obtained by Lindner, Mendelsohn and Sun in the following theorem. 1 (Lindner, Mendelsohn and Sun [47]). J(xy . yx = x ) contains precisely the set of all positive integers n = 0 or 1 (mod 4) except n = 5 , and possibly excepting n = 12 and 21. J. R. 2. There exists an idempotent Schroder quasigroup of order n for all positiue integers n = 0 or 1 (mod 4) except n = 5 and 9, and possibly excepting n = 12, 24, 33, 45, 69, 105, 117.

J. de Resmini, Steiner systems in finite projective planes, submitted. [4] T. Beth, D. Jungnickel and H . Lenz, Design theory (Cambridge University Press, Cambridge, New York, 1986). E. Block, On the orbits of collineation groups, Math. Zeif. 96 (1967) 33-49. [6] R. Brauer, On the connections between the ordinary and the modular characters of groups of finite order, Ann. Math. 42 (1941) 926-935. 171 F. Buekenhout, A. Delandtshecr and J . Doyen, Finite linear spaces with Hag-transitive groups, submitted.

There are models of the identity ( x y . x ) y = x in GF(4) for all prime powers 4 3 1 (mod 4). In particular, there are models of the identity of order n , where n E ( 5 , 9, 13, 17, 29, 49). 7, we readily obtain the following result. 3. J((xy * x ) y = x) contuins all positive integers v = 1 possibly v = 33, 57, 93, und 133. (mod 4), except It is still an open problem to determine more precisely J ( ( x y . x ) y = x). I t is not difficult to check that 2, 3, 4, and 6 do not belong to J ( ( x y x ) y = x).

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