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On transitive commutative idempotent quasigroups

Part of: Designs and configurations Other generalizations of groups

Published online by Cambridge University Press:  09 April 2009

Arnold Neumaier
Arnold Neumaier
Affiliation:
Fachbereich Mathematik Technische Universität D-1000 Berlin 12 West Germany
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Abstract

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Commutative idempotent quasigroups with a sharply transitive automorphism group G are described in terms of so-called Room maps of G. Orthogonal Room maps and skew Room maps are used to construct Room squares and skew Room squares. Very general direct and recursive constructions for skew Room maps lead to the existence of skew Room maps of groups of order prime to 30. Also some nonexistence results are proved.

MSC classification

Secondary: 05B15: Orthogonal arrays, Latin squares, Room squares 20N05: Loops, quasigroups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 27 , Issue 4 , June 1979 , pp. 411 - 429
Copyright
Copyright © Australian Mathematical Society 1979

References

Anderson, B. A. (1974), ‘A class of starter induced l-factorizations’, in Graphs and combinatorics (Lecture Notes in Mathematics 406, Springer, New York and Berlin), 180–185.CrossRefGoogle Scholar
Bruck, R. H. (1963), ‘ What is a loop? ’, Studies in modern algebra (Math. Association of America), 59–99.Google Scholar
Chong, B. C. (1972), Existence and construction of Room squares by the theory of loop and applications to experimental design (M.Sc. thesis, University of Singapore).Google Scholar
Gross, K. B. and Leonard, P. A. (1975), ‘Adders for the patterned starter in nonabelian groups’, J. Austral. Math. Soc. 21, 185–193.CrossRefGoogle Scholar
Mullin, R. C. and Nemeth, E. (1969), ‘An existence theorem for Room squares’, Canad. Math. Bull. 12, 493–497.CrossRefGoogle Scholar
Wallis, W. D. (1973a), ‘A Room square of side 257’, Proc. of the Fourth Southeastern Conference on Combinatorics, Graph Theory and Computing, 533.Google Scholar
Wallis, W. D. (1973b), ‘On one-factorizations of complete graphs’, J. Austral. Math. Soc. 16, 167–171.CrossRefGoogle Scholar
Wallis, W. D. (1974), ‘Solution of the Room square existence problem’, J. Combinatorial Theory A, 17, 379–383.CrossRefGoogle Scholar
Wallis, W. D. and Mullin, R. C. (1973), ‘Recent advances on complementary and skew Room squares’, Proc. of the Fourth Southeastern Conference on Combinatorics, Graph Theory and Computing, 521–532.Google Scholar
Wallis, W. D., Street, A. P. and Wallis, J. S. (1972), Combinatorics: Room squares, sum-free sets, Hadamard matrices (Lecture Notes in Mathematics 292, Springer, New York and Berlin).CrossRefGoogle Scholar