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Centraliser rings of multiplication groups on quasigroups

Published online by Cambridge University Press:  24 October 2008

J. D. H. Smith
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge

Extract

Let (Q,.) be a finite quasigroup, i.e. a finite set Q with a binary operation. called multiplication such that in the equation x.y = z any two elements determine the third uniquely. Then the mappings R(x): QQ; qq.x and L(x): QQ; qx.q are permutations of Q. The multiplication group G of Q is the subgroup of the symmetric group on Q generated by {R(x), L(x) | xQ}. If S is a field, G has a faithful representation Ḡ by permutation matrices acting on the S-vector space with Q as basis. The set of matrices commuting with Ḡ forms an S-algebra (under the usual operations) called the centraliser ring V(G, Q) of G on Q. The purpose of this note is to show how the permutation-theoretic object ‘centraliser ring’ may be expressed in terms of the quasigroup structure of Q, both to prepare one tool for the long-term programme of classifying finite quasigroups by means of their multiplication groups, and for comparison with the Schur ring method of group theory.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1976

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