Almost all quasigroups have rank 2
type: Item
instance of: scholarly article
title: Almost all quasigroups have rank 2
zbMATH Open document ID: 0786.20041
DOI: 10.1016/0012-365X(92)90537-P
author: Peter J. Cameron
published in: Discrete Mathematics
publication date: 1993-01-16T00:00:00Z
review text: The multiplication group \(\text{Mlt}(Q)\) of a quasigroup \(Q\) is the group of permutations of the underlying set of \(Q\) generated by the maps \(\lambda_ a: x\mapsto a\cdot x\) and \(\rho_ a: x\mapsto x\cdot a\). The question of whether \(\text{Mlt}(Q)\) is two-transitive is of importance in the study of the character theory of \(Q\). Here it is shown that almost all quasigroups have the property that the permutation group generated by merely the left mappings is either the symmetric group or the alternating group (almost all in the usual sense of the ratio of the number of quasigroups of size \(n\) with this property to the number of all quasigroups tending to 1 as \(n\) tends to \(\infty\)). The result is actually proved in terms of Latin squares; the multiplication table of a quasigroup is a Latin square and the group generated by the left mappings is just the group generated by the rows of the Latin square, regarded as permutations. The van der Waerden permanent conjecture implies that the number of Latin squares of order \(n\) is roughly \((n!)^ n/e^{n^ 2}\), whereas the number of Latin squares with row group distinct from \(S_ n\), \(A_ n\) is at most \(\sum| H|^ n\), where the summation is over the transitive subgroups \(H\) which are maximal in the sense that \(H < K < S_ n\) implies \(K = A_ n\). A straightforward argument considering the various possibilities for \(H\) establishes that this number is not too big. It is noteworthy that this result does not require the classification of the finite simple groups (though, of course, it does require another big gun, namely the proof of the van der Waerden permanent conjecture).
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 20B25
Mathematics Subject Classification ID: 05B15
Mathematics Subject Classification ID: 20B35
Mathematics Subject Classification ID: 20B20
Mathematics Subject Classification ID: 11N45
zbMATH DE Number: 94515
zbMATH Keywords: multiplication group
zbMATH Keywords: quasigroup
zbMATH Keywords: group of permutations
zbMATH Keywords: symmetric group
zbMATH Keywords: alternating group
zbMATH Keywords: number of quasigroups
zbMATH Keywords: multiplication table
zbMATH Keywords: van der Waerden permanent conjecture
zbMATH Keywords: number of Latin squares
reviewed by: Sheila Oates-Williams
MaRDI profile type: Publication
cites work: On the order of uniprimitive permutation groups
cites work: Finite Permutation Groups and Finite Simple Groups
cites work: The probability of generating the symmetric group
cites work: The solution of van der Waerden's problem for permanents
cites work: Proof of the van der Waerden conjecture regarding the permanent of a doubly stochastic matrix
cites work: Q5592263
cites work: Characters of finite quasigroups. III: Quotients and fusion
cites work: Notes on Egoritsjev's proof of the van der Waerden conjecture
cites work: Centraliser rings of multiplication groups on quasigroups
description: scientific article; zbMATH DE number 94515