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Loop transversals and the centralizer ring of a permutation group

Published online by Cambridge University Press:  24 October 2008

K. W. Johnson
K. W. Johnson
Affiliation:
Department of Mathematics, University of the West Indies, Kingston 7, Jamaica
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Extract

Many of the basic concepts in this paper are defined in (2), and it will be useful to the reader if he is familiar with that paper. In it the concept of a loop transversal to a subgroup S of a group G is defined and discussed, where if G is a transitive permutation group and no explicit S is mentioned it is assumed that S is a point stabilizer. In (3) there appeared the following problem (P):

Let G be a transitive permutation group which is 2-closed.

(i) Is there a fixed-point-free transversal in G (i.e. to a point stabilizer)?

(ii) Is there a loop transversal in G?

Unfortunately a note was added in which it was falsely stated that the answer to both (i) and (ii) is yes.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 94 , Issue 3 , November 1983 , pp. 411 - 416
Copyright
Copyright © Cambridge Philosophical Society 1983

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References

(1)Hall, M.Combinatorial Theory (Blaisdell, Massachusetts, 1967).Google Scholar
(2)Johnson, K. W.S-rings over loops, right mapping groups and transversals in permutation groups. Math. Proc. Cambridge Philos. Soc. 89 (1981), 433–443.CrossRefGoogle Scholar
(3)Johnson, K. W.Transversals, S-rings and centralizer rings of groups. Proceedings Algebra Carbondale 1980. Springer Lecture Notes, vol. 848, pp. 169–177.Google Scholar
(4)Wieland, H.Finite permutation groupa (Academic Press, New York, London, 1964).Google Scholar