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Bol loops of order pq

Published online by Cambridge University Press:  24 October 2008

Harald Niederreiter
Affiliation:
University of the West Indies, Kingston, Jamaica
Karl H. Robinson
Affiliation:
University of the West Indies, Kingston, Jamaica

Abstract

The main results of this paper show that for any odd primes p and q with q dividing p2 − 1 there exist non-associative Bol loops of order pq. The construction is sufficiently flexible so as to allow the resulting loops to be either Bruck loops or non-Bruck loops. It also permits the determination of isotopy classes among the loops constructed. All Bol loops of order 15 are determined.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1981

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References

REFERENCES

(1)Albert, A. A.Quasigroups II. Trans. Amer. Math. Soc. 55 (1944), 401419.CrossRefGoogle Scholar
(2)Bruck, R. H.A Survey of Binary Systems (Springer-Verlag, Berlin, 1958).CrossRefGoogle Scholar
(3)Burn, R. P.Finite Bol loops. Math. Proc. Cambridge Philos. Soc. 84 (1978), 377385.CrossRefGoogle Scholar
(4)Chein, O.Moufang loops of small order I. Trans. Amer. Math. Soc. 188 (1974), 3151.CrossRefGoogle Scholar
(5)Dickson, L. E.Linear Groups (Dover, New York, 1958).Google Scholar
(6)Glauberman, G.On loops of odd order I, II. J. Algebra 1 (1964), 374396;CrossRefGoogle Scholar
Glauberman, G.On loops of odd order I, II. J. Algebra 8 (1968), 393414.CrossRefGoogle Scholar
(7)Hall, M. JrThe Theory of Groups (Second Edition, Chelsea, New York, 1976).Google Scholar
(8)Leong, F.Moufang loops of order p 4. Nanta Math. 7 (1974), 3334.Google Scholar
(9)Mann, H. B.The construction of orthogonal latin squares. Ann. Math. Statistics 13 (1942), 418423.CrossRefGoogle Scholar
(10)Robinson, D. A.Some open questions on Bol loops, mimeographed notes. Oberwolfach Conference on Bol and Moufang Loops, 1976.Google Scholar
(11)Robinson, D. A.Bol loops. Trans. Amer. Math. Soc. 123 (1966), 341354.CrossRefGoogle Scholar
(12)Robinson, K. H.A note on Bol loops of order 2nk. Aequationes Math. (To appear).Google Scholar