I've been working on a classification of Bol loops of small order.

Here is *what I thought was* a complete list of the thirteen Bol loops of order 27.
Today (Sept 19, 2007) Michael Kinyon announced a couple new examples of exponent 3
with trivial centre! Now I need to go back and find a flaw in my program.
*Meanwhile the following list is not complete as it claims.*
And I haven't yet added Michael's new examples...

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 (25 KB) in plain text form.

Here are 13 (right) Bol loops of order 27, including 5 groups and 8 non-associative loops. These form 9 isotopy classes of such loops. The 8 nonassociative loops all have centre (equal to the centrum) of size 3; all have exponent nine; none are Moufang; and none are G-loops.

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). (This phenomenon does not occur, however, among
any the loops of order 27 in our list.)

- Two loops with 2 elements of order 3 and 24 elements of order 9: 27.2.3.0, 27.2.3.1
- Two loops with 8 elements of order 3 and 18 elements of order 9: 27.8.3.0, 27.8.3.1
- Two loops with 14 elements of order 3 and 12 elements of order 9: 27.14.3.0, 27.14.3.1
- Two loops with 20 elements of order 3 and 6 elements of order 9: 27.20.3.0, 27.20.3.1

The remaining four isotopy classes each consists of two isomorphism classes of non-associative loops. These four isotopy classes are the same as the four pairs listed above, i.e. the number of elements of order 3 is an isotopy invariant. (Contrast this with the situation for Bol loops of order 16, in which the number of involutions is by no means an isotopy invariant.)

As before, we 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.

/ revised June, 2002