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Quasiprimitivity and quasigroups

Published online by Cambridge University Press:  17 April 2009

J.D. Phillips  and
J.D.H. Smith
J.D. Phillips
Affiliation:
Department of MathematicsSaint Mary's CollegeMoraga, CA 94575, United States of America
J.D.H. Smith
Affiliation:
Department of MathematicsIowa State UniversityAmes, IA 50011, United States of America
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Abstract

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It is well known that Q is a simple quasigroup if and only if Mlt Q acts primitively on Q. Here we show that Q is a simple quasigroup if and only if Mlt Q acts quasiprimitively on Q, and that Q is a simple right quasigroup if and only if RMlt Q acts quasiprimitively on Q.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 59 , Issue 3 , June 1999 , pp. 473 - 475
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Bruck, R.H., A survey of binary systems, Ergebnisse der Mathematik 20 (Springer-Verlag, Berlin, Heidelberg, New York, 1958).CrossRefGoogle Scholar
[2]Chein, O., Pflugfelder, H.O. and Smith, J.D.H., editors, Quasigroups and loops, theory and applications, Sigma Series in Pure Mathematics 8 (Heldermann Verlag, Berlin, 1990).Google Scholar
[3]Johnson, K., Smith, J.D.H. and Song, S.Y., ‘Characters of finite quasigroups VI: Critical examples and doubletons’, European J. Combin. 11 (1990), 267–275.CrossRefGoogle Scholar
[4]Phillips, J.D., ‘A note on simple groups and simple loops’, in Conference proceedings on Groups – Korea (Pusan, 1998).Google Scholar
[5]Praeger, C.E., ‘An O'Nan-Scott theorem for finite quasiprimitive permutation groups and an application to 2–arc transitive graphs’, J. London Math. Soc. (2) 47 (1993), 227–239.CrossRefGoogle Scholar