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Sigma Series in Pure Mathematics -- Volume 7

   Enlarged Picture

Hala O. Pflugfelder

Quasigroups and Loops. An Introduction

160 pages, hard cover, ISBN 3-88538-007-2, EUR 35.00, 1990

This is a self-contained text written on an introductory level. Assuming as a prerequisite only an elementary course in abstract algebra, it develops from the beginnings the fairly young discipline of the theory of quasigroups. In geometry, it arose from the analysis of web structures; in algebra, from non-associative products; and in combinatorics, from Latin squares.



  Chapter I: Quasigroups and loops  
I.0 Introduction 1
I.1 Groupoids, quasigroups and loops 2
I.2 Subgroups and subloops 8
I.3 Nuclei and center of a quasigroup 16
I.4 Inverse property 20
I.5 Multiplication group and inner mapping group 24
I.6 Isomorphy 26
I.7 Homomorphy theory for quasigroups 28
  Chapter II: Quasigroups and geometry  
II.1 Quasigroups and 3-webs 34
II.2 Isotopy, parastrophy, isostrophy 39
II.3 Web configuations and loop laws 47
II.4 Affine incidence planes 52
  Chapter III: Isotopy theory for quasigroups  
III.1 Isotopy for groupoids 57
III.2 Isotopy theory for quasigroups and loops 59
III.3 Autotopisms 66
III.4 Pseudo-automorphisms of quasigroups 74
III.5 Derivatives 79
III.6 The isotopy-isomorphy property of loops 82
  Chapter IV: Moufang loops  
IV.1 Basic properties of Moufang loops 88
IV.2 Moufang's theorem 93
IV.3 Some theorems concerning pseudo-automorphisms, the nucleus, the Moufang center and self-adjoint subgroups of Moufang loops 97
IV.4 Isotopy of Moufang loops 101
IV.5 Commutative Moufang loops 107
IV.6 Bol loops 112
  Chapter V: Some classes of quasigroups  
V.1 Totally symmetric quasigroups 122
V.2 Distributive and entropic quasigroups 131
  Bibliography 143
  Subject index 146