Bol Loops of Small Order
I have been working on classifying Bol loops of small order, using a backtrack program described
below. This program easily finds that there are 11 Bol loops of order 8,
3 of order 15, etc. (including groups), in agreement with known results. The classification of 11 Bol
loops of order 8, for example, is completed in 3.2 seconds on my laptop computer (with a Pentium III
at 700 MHz).
Currently I have completed the classification of Bol loops of orders
16, 21 and
27.
I intend to provide links here to classifications of Bol loops of other small orders as
they are completed. I have also provided a crossreference list
comparing the names of Bol loops, as they appear in other sources, with the names in my list.
I would appreciate an email
message () from you if you have any comments regarding this list.
Bol Loops of Order Less Than 32
In the following table we omit loops of order p, 2p or p^{2} (for p prime),
which by the results of Burn, must be groups.
For a list including all groups of order less than 32 see our
crossreference list.
n  Bol Loops of order n
 NonAssociative Bol Loops of order n
 Groups of order n
 NonAssociative Moufang Loops of order n


8
 11
 6
 5
 0


12
 8
 3
 5
 1


15
 3
 2
 1
 0


16
 2052
 2038
 14
 5


18
 6
 1
 5
 0


20
 8
 3
 5
 1


21
 4
 2
 2
 0


24
 ≥103
 ≥88
 15
 5


27
 13
 8
 5
 0


28
 7
 3
 4
 1


30
 ≥6
 ≥2
 4
 0


References
 Hala O. Pflugfelder, Quasigroups and Loops: Introduction, Heldermann Verlag, Berlin, 1990.
 George Glauberman, On Loops of Odd Order II, Journal of Algebra 8 (1968), 393414.
 Edgar G. Goodaire and Sean May,
Bol Loops of Order less than 32,
Dept. of Math and Statistics, Memorial University of Newfoundland, Canada, 1995.
(List of Bol loops of order n<32; exhaustive for n not equal to 16, 21, 24, 27, 30.)
 R.P. Burn, Finite Bol loops, Math. Proc. Cambridge Philos. Soc. 84 (1978), 377385.
(Burn shows that any Bol loop of order 2p or p^{2}, where p is prime,
is necessarily a group; and he lists all Bol loops of order 8.)
 R.P. Burn, Finite Bol loops II, Math. Proc. Cambridge Philos. Soc. 88 (1981), 445455.
(Shows that there are exactly two nonassociative, nonMoufang Bol loops of order 4p for any odd
prime p.)
 R.P. Burn, Finite Bol loops III, Math. Proc. Cambridge Philos. Soc. 97 (1985), 219223.
(Shows the uniqueness of the nonassociative Bol loop of order 2p^{2} for any odd
prime p.)
 Edgar G. Goodaire, Sean May and Maitreyi Raman,
The Moufang Loops of Order less than 64, Nova Science, 1999.
 H. Niederreiter and K.H. Robinson, Bol loops of order pq, Math. Proc. Cambridge Philos.
Soc. 89 (1981), 241256.
(Shows that there are just 2 Bol loops of order 15.)
Definitions
Here is a quick review of the relevant definitions; for further explanation, proofs, etc., please consult
references such as those listed above.
A loop is a set L
with binary operation (denoted here simply by juxtaposition) such that
 for each a in L, the left multiplication map
L_{a}:L−>L, x−>ax is bijective;
 for each a in L, the right multiplication map
R_{a}:L−>L, x−>xa is bijective; and
 L has a twosided identity 1.
The order of L is its cardinality L.
A loop L is a (right) Bol loop if
(xy.z)y=x(yz.y) for all x,y,z in L.
Bol loops are power associative, i.e. for all a in L
and every integer k, the power a^{k} is welldefined (independent of
order in which the multiplications are performed). The order of an element
a is the smallest positive integer k for which a^{k}=1.
The order of any element of a finite Bol loop must divide the order of the loop.
The exponent of a finite Bol loop L is the smallest positive integer k such that
a^{k}=1 for all a in L; note that the exponent equals the least common
multiple of a for a in L, which divides L.
Let I(L) denote the set of involutions (elements of order 2) in the Bol loop L.
A loop is Left Bol if it satisfies the left Bol identity
(x.yx)z=x(y.xz) for all x,y,z in L. A loop is Moufang if it is both left Bol
and (right) Bol.
The centrum of L is the set C(L) of all elements of L which commute
with every element of L. The centre of L is the set Z(L) of all elements of
L which both commute and associate with every element of L.
The left (resp. right; full) multiplication group of L is the
group of permutations of L generated by all left (resp. right;
both left and right) multiplication maps.
The inner mappings are the elements of the subgroup Inn(L) of the multiplication group
of L, generated by the permutations
T_{a}=R_{a}L_{a}^{−1},
R_{a,b}=R_{a}R_{b}R_{ab}^{−1},
L_{a,b}=L_{a}L_{b}L_{ba}^{−1} (with lefttoright composition
of permutations), all of which fix 1. A conjugacy class of L is an orbit of Inn(L).
A loop has property A_{l} (resp. A_{r}) if every left inner mapping L_{a,b}
(resp. right inner mapping R_{a,b})is an automorphism of the loop.
A loop L has the automorphic inverse property if
x^{−1}y^{−1}=(xy)^{−1} for all x,y in L.
Thanks to Matteo Allegro for pointing out (April, 2012) an error in the automated checks for the automorphic inverse property
in this catalogue of Bol loops; I hope to have this correced sometime soon.
For any elements x,y,z of a loop L, we define the commutator (x,y) by
xy=yx.(x,y); and the associator (x,y,z) by
xy.z=(x.yz)(x,y,z). The commutator (resp. associator)
subloop of L is the subloop generated by all commutators (resp. associators) in L.
Methodology Used
Here we outline the algorithm used in our exhaustive search for representatives of all isomorphism classes of
Bol loops of order n.
Define a partial Bol loop of order n and size k to be a set S consisting of k permutations of
{0,1,2,...,n−1} such that
 (PB1) the identity permutation belongs to S;
 (PB2) whenever g and h are distinct elements of S, the permutation gh^{−1}
is fixedpointfree;
 (PB3) whenever g and h are in S, so is ghg.
Note that a Bol loop L of order n is equivalent to a partial Bol loop S of order n
and size k=n, in which S consists of the columns of the multiplication table of L.
Our basic tool is a program which accepts as input (i) a partial Bol loop of order n and size k<n,
and (ii) a set D consisting of distinguished divisors of n. This program looks for all ways to
 choose a
permutation s whose order lies in D, such that the new set S' generated by the union of S
and {s} subject to the operation (PB3) above, satisfies (PB2) and thus forms a partial Bol loop
of size k'>k; then
 repeat until a Bol loop of order n is obtained.
This is accomplished using a backtrack search, and each Bol loop produced is tested for isomorphism with previously
stored loops; if it is in fact new, then it is stored, and otherwise it is not stored. In order to produce
canonical representatives of isomorphism classes of Bol loops for the purpose of performing this isomorphism test,
we first encode the algebraic content of each Bol loop as a graph, then using Brendan McKay's software package
nauty to find a "canonical" representative for each
of the resulting graphs. Moreover we reduce the size of the search tree significantly by rejecting isomorphic copies
not only of the completed Bol loops, but also of partial Bol loops, or even of multiplication tables with incomplete
columns. However, due to the expense (in terms of execution time) of producing these canonical isomorphs,
we perform this isomorphism rejection only at nodes of the search tree close to the root.
We have also made use of the general theory to reduce the search space. For example, to search for all Bol loops of
order 15, we may start with the partial Bol loop of order 5 generated
by (0,1,2,3,4)(5,6,7,8,9)(10,11,12,13,14), and set D={3,5} for the possible orders of elements to be adjoined.
This is because any Bol loop of order 15 must have an element of order 5 (by Theorem 15 of Glauberman's paper
cited above).
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 March, 2007