Bol Loops of Order 27
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
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.)
8 non-associative loops, listed by number of elements of order 3
- Two loops with 2 elements of order 3 and 24 elements of order 9: 184.108.40.206,
- Two loops with 8 elements of order 3 and 18 elements of order 9: 220.127.116.11,
- Two loops with 14 elements of order 3 and 12 elements of order 9:
- Two loops with 20 elements of order 3 and 6 elements of order 9: 18.104.22.168,
9 Isotopy ClassesThere are five groups of order 27: 22.214.171.124
(abelian C3×C9), 126.96.36.199
(elementary abelian), 188.8.131.52
(nonabelian of exponent 3), 184.108.40.206
(nonabelian of exponent 9).
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.)
Naming of the LoopsFor each of the loops of order 27, I have used a
name 27.t.c.k where t is the number of elements of
L of order three, 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.
I have searched for Bol loops of order 27 by backtrack search as
described elsewhere. However in this particular
case the permutations comprising the partial Bol loops through which we must search
(columns of the Cayley table) are chosen not from the full symmetric group of degree 27,
but from a Sylow 3-subgroup thereof, a subgroup of size "only" 1594323 (see Glauberman 1968, Theorem 15). This reduces
the search space considerably.
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.
AcknowledgementI am grateful to the Department of Mathematics and Statistics, Memorial University of Newfoundland for their
hospitality while much of this study was being undertaken, and in particular to Edgar Goodaire for conversations which
have stimulated me in this direction.