Bol Loops of Order 8

I've been working on a classification of Bol loops of small order. Here is a list of the 11 Bol loops of order 8. This includes the 5 groups of order 8 and the 6 non-associative Bol loops of order 8 (none of which are Moufang). The completeness of this list was first shown by Burn (1978), but I have independently verified this list using my own program. I would appreciate an email message () from you if you have any comments regarding this work.

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 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 8 in our list.)

The 11 Bol loops of order 8, tabulated by number of involutions and size of centrum

	\|C(L)\|=2 (5 loops)	\|C(L)\|=4 (3 loops)	\|C(L)\|=8 (3 loops)
\|I(L)\|=1 (3 loops)	8.1.2.0	8.1.4.0	8.1.8.0
\|I(L)\|=3 (3 loops)	8.3.2.0, 8.3.2.1		8.3.8.0
\|I(L)\|=5 (3 loops)	8.5.2.0	8.5.4.0, 8.5.4.1
\|I(L)\|=7 (2 loops)	8.7.2.0		8.7.8.0

The 7 Isotopy Classes of Bol Loops

The 5 Groups

Isotopy Classes 0,1,2,3,4 8.1.2.0, 8.1.8.0, 8.3.8.0, 8.5.2.0, 8.7.8.0

The 6 non-associative Bol loops (non-Moufang, non-G-loops)

Isotopy Class 5 8.1.4.0, 8.3.2.1, 8.5.4.1

Isotopy Class 6 8.3.2.0, 8.5.4.0, 8.7.2.0

Naming of the Loops

For each of the loops of order 8, I have used a name 8.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.

Methodology Used

I have enumerated the Bol loops of order 8 by exhaustive backtrack search as described elsewhere. I have retained a single representative of each isomorphism class of the resulting loops. To accomplish this, we have represented L by a graph which encodes all information in the Cayley table of L. I have then used Brendan McKay's software package nauty to find a "canonical" representative for each resulting graph.

I have also used the computational algebra package GAP (Graphs, Algorithms and Programming) to compute orders of left, right and full multiplication groups.

Acknowledgement

I am grateful to the Department of Mathematics and Statistics, Memorial University of Newfoundland for their hospitality while this study is being undertaken, and in particular to Edgar Goodaire for conversations which have stimulated me in this direction.

/ revised July, 2003