Permutation representations of loops.
type: Item
instance of: scholarly article
title: Permutation representations of loops.
zbMATH Open document ID: 1033.20078
DOI: 10.1016/S0021-8693(03)00179-0
author: J. D. H. Smith
published in: Journal of Algebra
publication date: 2003-07-30T00:00:00Z
review text: The author initiates a theory of permutation representations of finite loops. Such a theory can be introduced more generally for a finite quasigroup \(Q\) by considering, for a given subquasigroup \(P\), the elements of the corresponding homogeneous space \(P\setminus Q\) as orbits on \(Q\) of the group of permutations generated by the left multiplications by elements of \(P\). Each element of \(Q\) yields a Markov chain action on the homogeneous space \(P\setminus Q\) as a set of states. If \(P\) is a subgroup of a group \(Q\) this concept specializes to the usual one of a homogeneous space or transitive permutation representation for groups. For finite loops the author obtains that the class of permutation representations is closed under disjoint unions and direct products, each representation decomposing into a disjoint union of irreducible representations. But, in contrast with the group case, he shows that a loop need not to be recoverable to within isomorphism from a faithful permutation representation. The final section is devoted to an interesting application of this theory to the study of the so-called `Lagrange properties' of a finite loop.
reviewed by: Elena Zizioli
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 20C15
zbMATH DE Number: 1956738
zbMATH Keywords: finite loops
zbMATH Keywords: finite quasigroups
zbMATH Keywords: permutation representations
zbMATH Keywords: Lagrange properties
zbMATH Keywords: iterated function systems
zbMATH Keywords: representations of loops
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1956738