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Latin squares with highly transitive automorphism groups

Part of: Designs and configurations Permutation groups

Published online by Cambridge University Press:  09 April 2009

R. A. Bailey
R. A. Bailey
Affiliation:
Mathematics Faculty The Open UniversityMilton Keynes, U.K.
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Abstract

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A Latin square is considered to be a set of n2 cells with three block systems. An automorphisni is a permutation of the cells which preserves each block system. The automorphism group of a Latin Square necessarily has at least 4 orbits on unordered pairs of cells if n < 2. It is shown that there are exactly 4 orbits if and only if the square is the composition table of an elementary abelian 2-group or the cclic group of order 3.

MSC classification

Secondary: 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures 05B15: Orthogonal arrays, Latin squares, Room squares
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 33 , Issue 1 , August 1982 , pp. 18 - 22
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Denès, J. and Keedwell, A. D., Latin squares and their applications (English Universities Press, London, 1974).Google Scholar
[2]Heinrich, K. and Wallis, W. D., ‘The maximum number of intercalates in a Latin square’, Proceedings of the Eighth Australian Conference on Combinatorial Mathematics (1981).CrossRefGoogle Scholar
[3]Sade, A., ‘Quasigroupes isotopes. Autotopies d'un groupe’, Ann. Soc. Sci. Bruxelles Sér. 181 (1967), 231–239.Google Scholar
[4]Schönhardt, E., ‘Über lateinische Quadrate und Unionen’, J. Reine Angew. Math. 163 (1930), 183–229.CrossRefGoogle Scholar
[5]Preece, D. A., Bailey, R. A. and Patterson, H. D., ‘A randomization problem in forming designs with superimposed treatments’, Austral. J. Statist. 20 (1978), 111–125.CrossRefGoogle Scholar