Bol Loops of Order 12

I've been working on a classification of Bol loops of small order. Here is a list of the eight Bol loops of order 12. This includes the five groups of order 12, the single non-associative Moufang loop of order 12 and the two non-Moufang Bol loops of order 12. The completeness of this list was first shown by Burn (1981). I would appreciate an email message () from you if you have any comments regarding this work.

I have made available

In listing elements of the commutator (resp. associator) subloop of each of our loops, we have printed in italics any elements which are not actual commutators (resp. associators). (I haven't checked, however, whether in fact this phenomenon occurs among any the loops of order 12 in our list.)

The eight Bol loops of order 12, tabulated by number of involutions and size of centrum

|C(L)|=1   (3 loops) |C(L)|=2   (2 loops) |C(L)|=3   (1 loop) |C(L)|=12   (2 loops)
|I(L)|=1   (2 loops)   12.1.2.0   12.1.12.0
|I(L)|=3   (2 loops) 12.3.1.0     12.3.12.0
|I(L)|=5   (1 loop)     12.5.3.0  
|I(L)|=7   (1 loop)   12.7.2.0    
|I(L)|=9   (2 loops) 12.9.1.0, 12.9.1.1      

The 7 Isotopy Classes of Bol Loops

The five groups: Isotopy Classes 0,1,2,3,4   12.1.12.0,   12.1.2.0,   12.3.1.0,   12.3.12.0,   12.7.2.0
The non-associative Moufang loop: Isotopy Class 5   12.9.1.1
The non-Moufang Bol loops: Isotopy Class 6   12.5.3.0,   12.9.1.0

Naming of the Loops

For each of the loops of order 12, I have used a name 12.i.c.k where i=|I(L)|, c=|C(L)| and the index k=0,1,2,... indicates merely the order in which each isomorphism class of loop was first encountered by my computer.

/ revised July, 2003