See here for general background on loops.
A loop Q is a (right) Bol loop if it satisfies the right Bol identity ((xy)z)y = x((yz)y.
One of the open problems in the theory of Bol loops is whether or not every finite Bol p-loop (that is, a Bol loop of order a power of a prime p) has nontrivial center. For groups, this is well-known to be true. For Bruck p-loops with p odd (Bruck loops are Bol loops with the automorphic inverse property), the result is due to Glauberman [G1]. Glauberman also proved the result for Moufang p-loops with p odd [G2]. For Moufang 2-loops, the result is due to Glauberman and Wright [GW].
For Bol loops, the case p = 2 was resolved in the negative by Kiechle and Nagy [KN]. They classified all Bol loops of order 16 and exponent 2 with trivial center.
It has been generally believed that for odd primes p, finite Bol p-loops have nontrivial center, and in fact, are centrally nilpotent. For instance, Eric Moorhouse's list of known Bol loops of order 27 (at one time thought to be a classification) certainly suggested this for Bol 3-loops. In the odd case, a Bol p-loop is solvable. (Sketch of proof: It is sufficient to show that such a loop cannot be simple. The right multiplication group G is a p-group, as follows from a theorem of Fischer [F] (or a theorem in [FKP]). The right inner mapping group is a proper, nonnormal subgroup of G of index at least p3. (Bol loops of orders p and p2 are groups [B].) Its normal closure, say H, is then a proper subgroup of G, of index at least p. The orbit of H through the neutral element is then a proper normal subloop of index at least p.)
Here I present two Bol loops of order 27 with trivial center.
I'd like to thank Gabor Nagy, who encouraged me to revisit the question and for the argument above. Gabor solved the main open problem in the theory of Bol loops (and in loop theory in general) with his constructions of finite, simple, proper (nonMoufang) Bol loops. I'd also like to thank my colleague Petr Vojtechovsky; it's a great privilege to be at an institution with another loop theorist, especially one as talented as Petr. Gabor and Petr developed the LOOPS package for the computer algebra system GAP, which I used heavily to investigate these loops.
I thank Tuval Foguel, who will be coauthor of the paper in which all of this will eventually appear. Ales Drapal has been an inspiration to me. Finally, long-time friend and correspondent J. D. Phillips was a coauthor with Tuval and myself of some predecessor work on Bol loops of odd order [FKP].
These loops are isotopic, and together form a complete isotopism class. Each loop has exponent 3. Being isotopic, they have the same right multiplication group; it has order 243 (GAP identifies it as group [243,37].) The right inner mapping group is an elementary abelian group of order 9.
The full multiplication group (which coincides with the left multiplication group) has order 26 37. (If the order of the multiplication group were a power of 3, then it follows from a theorem of A. A. Albert that the centers of the loops would be nontrivial [A].)
Not only do the loops have trivial center, they also have trivial right (= middle) nucleus. The commutant (also known as the centrum, semicenter, or commutative center), which is the set of those elements which commute with all elements, is also trivial.
Each loop has a unique nontrivial normal subgroup (associative subloop), which is elementary abelian of order 9. This subgroup is the left nucleus, the derived subloop, and the associator subloop.
The Cayley tables of the loops follow:
[ [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27 ], [ 2, 3, 1, 5, 6, 4, 8, 9, 7, 15, 13, 14, 18, 16, 17, 12, 10, 11, 22, 23, 24, 25, 26, 27, 19, 20, 21 ], [ 3, 1, 2, 6, 4, 5, 9, 7, 8, 17, 18, 16, 11, 12, 10, 14, 15, 13, 25, 26, 27, 19, 20, 21, 22, 23, 24 ], [ 4, 5, 6, 7, 8, 9, 1, 2, 3, 14, 15, 13, 17, 18, 16, 11, 12, 10, 26, 27, 25, 20, 21, 19, 23, 24, 22 ], [ 5, 6, 4, 8, 9, 7, 2, 3, 1, 16, 17, 18, 10, 11, 12, 13, 14, 15, 20, 21, 19, 23, 24, 22, 26, 27, 25 ], [ 6, 4, 5, 9, 7, 8, 3, 1, 2, 12, 10, 11, 15, 13, 14, 18, 16, 17, 23, 24, 22, 26, 27, 25, 20, 21, 19 ], [ 7, 8, 9, 1, 2, 3, 4, 5, 6, 18, 16, 17, 12, 10, 11, 15, 13, 14, 24, 22, 23, 27, 25, 26, 21, 19, 20 ], [ 8, 9, 7, 2, 3, 1, 5, 6, 4, 11, 12, 10, 14, 15, 13, 17, 18, 16, 27, 25, 26, 21, 19, 20, 24, 22, 23 ], [ 9, 7, 8, 3, 1, 2, 6, 4, 5, 13, 14, 15, 16, 17, 18, 10, 11, 12, 21, 19, 20, 24, 22, 23, 27, 25, 26 ], [ 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 20, 25, 27, 26, 22, 24, 23, 1, 3, 2, 7, 9, 8, 4, 6, 5 ], [ 11, 12, 10, 14, 15, 13, 17, 18, 16, 27, 26, 25, 24, 23, 22, 21, 20, 19, 8, 7, 9, 5, 4, 6, 2, 1, 3 ], [ 12, 10, 11, 15, 13, 14, 18, 16, 17, 23, 22, 24, 20, 19, 21, 26, 25, 27, 6, 5, 4, 3, 2, 1, 9, 8, 7 ], [ 13, 14, 15, 16, 17, 18, 10, 11, 12, 21, 20, 19, 27, 26, 25, 24, 23, 22, 9, 8, 7, 6, 5, 4, 3, 2, 1 ], [ 14, 15, 13, 17, 18, 16, 11, 12, 10, 26, 25, 27, 23, 22, 24, 20, 19, 21, 4, 6, 5, 1, 3, 2, 7, 9, 8 ], [ 15, 13, 14, 18, 16, 17, 12, 10, 11, 22, 24, 23, 19, 21, 20, 25, 27, 26, 2, 1, 3, 8, 7, 9, 5, 4, 6 ], [ 16, 17, 18, 10, 11, 12, 13, 14, 15, 20, 19, 21, 26, 25, 27, 23, 22, 24, 5, 4, 6, 2, 1, 3, 8, 7, 9 ], [ 17, 18, 16, 11, 12, 10, 14, 15, 13, 25, 27, 26, 22, 24, 23, 19, 21, 20, 3, 2, 1, 9, 8, 7, 6, 5, 4 ], [ 18, 16, 17, 12, 10, 11, 15, 13, 14, 24, 23, 22, 21, 20, 19, 27, 26, 25, 7, 9, 8, 4, 6, 5, 1, 3, 2 ], [ 19, 20, 21, 22, 23, 24, 25, 26, 27, 1, 7, 4, 6, 3, 9, 8, 5, 2, 10, 18, 14, 12, 17, 13, 11, 16, 15 ], [ 20, 21, 19, 23, 24, 22, 26, 27, 25, 5, 2, 8, 7, 4, 1, 3, 9, 6, 16, 15, 11, 18, 14, 10, 17, 13, 12 ], [ 21, 19, 20, 24, 22, 23, 27, 25, 26, 9, 6, 3, 2, 8, 5, 4, 1, 7, 13, 12, 17, 15, 11, 16, 14, 10, 18 ], [ 22, 23, 24, 25, 26, 27, 19, 20, 21, 2, 8, 5, 4, 1, 7, 9, 6, 3, 15, 11, 16, 14, 10, 18, 13, 12, 17 ], [ 23, 24, 22, 26, 27, 25, 20, 21, 19, 6, 3, 9, 8, 5, 2, 1, 7, 4, 12, 17, 13, 11, 16, 15, 10, 18, 14 ], [ 24, 22, 23, 27, 25, 26, 21, 19, 20, 7, 4, 1, 3, 9, 6, 5, 2, 8, 18, 14, 10, 17, 13, 12, 16, 15, 11 ], [ 25, 26, 27, 19, 20, 21, 22, 23, 24, 3, 9, 6, 5, 2, 8, 7, 4, 1, 17, 13, 12, 16, 15, 11, 18, 14, 10 ], [ 26, 27, 25, 20, 21, 19, 23, 24, 22, 4, 1, 7, 9, 6, 3, 2, 8, 5, 14, 10, 18, 13, 12, 17, 15, 11, 16 ], [ 27, 25, 26, 21, 19, 20, 24, 22, 23, 8, 5, 2, 1, 7, 4, 6, 3, 9, 11, 16, 15, 10, 18, 14, 12, 17, 13 ] ]
[ [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27 ], [ 2, 3, 1, 5, 6, 4, 8, 9, 7, 15, 13, 14, 18, 16, 17, 12, 10, 11, 22, 23, 24, 25, 26, 27, 19, 20, 21 ], [ 3, 1, 2, 6, 4, 5, 9, 7, 8, 17, 18, 16, 11, 12, 10, 14, 15, 13, 25, 26, 27, 19, 20, 21, 22, 23, 24 ], [ 4, 5, 6, 7, 8, 9, 1, 2, 3, 14, 15, 13, 17, 18, 16, 11, 12, 10, 26, 27, 25, 20, 21, 19, 23, 24, 22 ], [ 5, 6, 4, 8, 9, 7, 2, 3, 1, 16, 17, 18, 10, 11, 12, 13, 14, 15, 20, 21, 19, 23, 24, 22, 26, 27, 25 ], [ 6, 4, 5, 9, 7, 8, 3, 1, 2, 12, 10, 11, 15, 13, 14, 18, 16, 17, 23, 24, 22, 26, 27, 25, 20, 21, 19 ], [ 7, 8, 9, 1, 2, 3, 4, 5, 6, 18, 16, 17, 12, 10, 11, 15, 13, 14, 24, 22, 23, 27, 25, 26, 21, 19, 20 ], [ 8, 9, 7, 2, 3, 1, 5, 6, 4, 11, 12, 10, 14, 15, 13, 17, 18, 16, 27, 25, 26, 21, 19, 20, 24, 22, 23 ], [ 9, 7, 8, 3, 1, 2, 6, 4, 5, 13, 14, 15, 16, 17, 18, 10, 11, 12, 21, 19, 20, 24, 22, 23, 27, 25, 26 ], [ 10, 14, 18, 11, 15, 16, 12, 13, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 1, 4, 7, 8, 2, 5, 6, 9, 3 ], [ 11, 15, 16, 12, 13, 17, 10, 14, 18, 27, 25, 26, 21, 19, 20, 24, 22, 23, 8, 2, 5, 6, 9, 3, 1, 4, 7 ], [ 12, 13, 17, 10, 14, 18, 11, 15, 16, 23, 24, 22, 26, 27, 25, 20, 21, 19, 6, 9, 3, 1, 4, 7, 8, 2, 5 ], [ 13, 17, 12, 14, 18, 10, 15, 16, 11, 21, 19, 20, 24, 22, 23, 27, 25, 26, 9, 3, 6, 4, 7, 1, 2, 5, 8 ], [ 14, 18, 10, 15, 16, 11, 13, 17, 12, 26, 27, 25, 20, 21, 19, 23, 24, 22, 4, 7, 1, 2, 5, 8, 9, 3, 6 ], [ 15, 16, 11, 13, 17, 12, 14, 18, 10, 22, 23, 24, 25, 26, 27, 19, 20, 21, 2, 5, 8, 9, 3, 6, 4, 7, 1 ], [ 16, 11, 15, 17, 12, 13, 18, 10, 14, 20, 21, 19, 23, 24, 22, 26, 27, 25, 5, 8, 2, 3, 6, 9, 7, 1, 4 ], [ 17, 12, 13, 18, 10, 14, 16, 11, 15, 25, 26, 27, 19, 20, 21, 22, 23, 24, 3, 6, 9, 7, 1, 4, 5, 8, 2 ], [ 18, 10, 14, 16, 11, 15, 17, 12, 13, 24, 22, 23, 27, 25, 26, 21, 19, 20, 7, 1, 4, 5, 8, 2, 3, 6, 9 ], [ 19, 21, 20, 25, 27, 26, 22, 24, 23, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11, 16, 18, 17, 13, 15, 14 ], [ 20, 19, 21, 26, 25, 27, 23, 22, 24, 5, 6, 4, 8, 9, 7, 2, 3, 1, 16, 18, 17, 13, 15, 14, 10, 12, 11 ], [ 21, 20, 19, 27, 26, 25, 24, 23, 22, 9, 7, 8, 3, 1, 2, 6, 4, 5, 13, 15, 14, 10, 12, 11, 16, 18, 17 ], [ 22, 24, 23, 19, 21, 20, 25, 27, 26, 2, 3, 1, 5, 6, 4, 8, 9, 7, 15, 14, 13, 12, 11, 10, 18, 17, 16 ], [ 23, 22, 24, 20, 19, 21, 26, 25, 27, 6, 4, 5, 9, 7, 8, 3, 1, 2, 12, 11, 10, 18, 17, 16, 15, 14, 13 ], [ 24, 23, 22, 21, 20, 19, 27, 26, 25, 7, 8, 9, 1, 2, 3, 4, 5, 6, 18, 17, 16, 15, 14, 13, 12, 11, 10 ], [ 25, 27, 26, 22, 24, 23, 19, 21, 20, 3, 1, 2, 6, 4, 5, 9, 7, 8, 17, 16, 18, 14, 13, 15, 11, 10, 12 ], [ 26, 25, 27, 23, 22, 24, 20, 19, 21, 4, 5, 6, 7, 8, 9, 1, 2, 3, 14, 13, 15, 11, 10, 12, 17, 16, 18 ], [ 27, 26, 25, 24, 23, 22, 21, 20, 19, 8, 9, 7, 2, 3, 1, 5, 6, 4, 11, 10, 12, 17, 16, 18, 14, 13, 15 ] ]