
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 3webs 
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 
Pseudoautomorphisms of quasigroups 
74 
III.5 
Derivatives 
79 
III.6 
The isotopyisomorphy 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 pseudoautomorphisms, the nucleus, the Moufang center
and selfadjoint 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 