Bol Loops of Order 18

I've been working on a classification of Bol loops of small order. Here is a list of the 6 Bol loops of order 18. This includes the five groups of order 18 and the single non-Moufang (and non-associative) Bol loop of order 18. No two of these examples are isotopic. The completeness of this list follows from Burn (1985).

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 (5 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). (This phenomenon occurs in the case of loop 18.3.3.0.)

The 6 Bol Loops

The Cyclic Group 18.1.18.0

The Group C₃ × C₆ 18.1.18.1

The Dihedral Group 18.9.1.1

The Group S₃ × C₃ 18.3.3.1

The Group C₃² : C₂ 18.9.1.0

The non-Moufang (non-associative) Bol Loop 18.3.3.0

Naming of the Loops

For each of the loops of order 18, I have used a name 18.i.c.k where i is the number of involutions, 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 February, 2005