Bol Loops of Order 15

I've been working on a classification of Bol loops of small order. Here is a list of the 3 Bol loops of order 15. This includes the cyclic group of order 15 and the 2 non-associative Bol loops of order 15 (neither of which is Moufang). The two non-associative examples are isotopic to one another. The completeness of this list was first shown by Niederreiter and Robinson (1981).

I have made available

a quick review of the relevant definitions;
a suitable list of references for further explanation, proofs, etc.;
a cross-reference list comparing this list with previously published lists of groups and loops;
descriptions of these loops in html format, found through the links below; and
the Cayley tables (2 KB) in plain text form.

In listing elements of the commutator (resp. associator) subloop of each of these 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 15 in our list.)

The 2 Isotopy Classes of Bol Loops

Isotopy Class 0: The Cyclic Group 15.2.15.0

Isotopy Class 1: The 2 Non-Moufang (non-associative) Bol Loops 15.10.1.0, 15.10.1.1

Naming of the Loops

For each of the loops of order 15, I have used a name 15.i.c.k where i is the number of 3-elements, 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 March, 2004