Any kind of remarks, comments and questions are welcome!
The set L endowed by the binary operation x·y=xy is a quasigroup if the equation xy=z has a unique solution for x and y. Quasigroups with a neutral element 1 are called loops. The loop L is a (left) Bol loop if it satisfies the left Bol identity x(y(xz)) = (x(yx))z.
The existence of finite simple proper Bol loops was considered as one of the main problems of the theory of loops and quasigroups.
On this page we present three finite simple proper left Bol loops. The first two loops are results of a more general construction which is explained in this preprint. In particular, these two loops are G-loops.
These loops were constructed and investigated using the computer algebra system GAP and the GAP package LOOPS.
The results were obtained during my Marie Curie Fellowship MEIF-CT-2006-041105 at the University of Würzburg (Germany).
I would like to thank Michael Kinyon, Peter Müller and Petr Vojtěchovský for many stimulating conversations and helpful comments.
This loop has a solvable left multiplication group of order 288. The left and middle nuclei are trivial, but the right nucleus has order 2.
The Cayley table of this loop is the following:
[ [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24 ], [ 2, 1, 5, 6, 3, 4, 14, 13, 17, 18, 15, 16, 8, 7, 11, 12, 9, 10, 20, 19, 23, 24, 21, 22 ], [ 3, 4, 1, 2, 6, 5, 9, 10, 7, 8, 12, 11, 21, 22, 19, 20, 24, 23, 15, 16, 13, 14, 18, 17 ], [ 4, 3, 6, 5, 1, 2, 16, 15, 18, 17, 13, 14, 22, 21, 24, 23, 19, 20, 10, 9, 12, 11, 7, 8 ], [ 5, 6, 2, 1, 4, 3, 23, 24, 20, 19, 22, 21, 11, 12, 8, 7, 10, 9, 17, 18, 14, 13, 16, 15 ], [ 6, 5, 4, 3, 2, 1, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7 ], [ 7, 8, 12, 11, 9, 10, 19, 20, 24, 23, 21, 22, 1, 2, 6, 5, 3, 4, 13, 14, 18, 17, 15, 16 ], [ 8, 7, 10, 9, 11, 12, 2, 1, 4, 3, 5, 6, 20, 19, 22, 21, 23, 24, 14, 13, 16, 15, 17, 18 ], [ 9, 10, 11, 12, 7, 8, 21, 22, 23, 24, 19, 20, 15, 16, 17, 18, 13, 14, 3, 4, 5, 6, 1, 2 ], [ 10, 9, 8, 7, 12, 11, 4, 3, 2, 1, 6, 5, 16, 15, 14, 13, 18, 17, 22, 21, 20, 19, 24, 23 ], [ 11, 12, 9, 10, 8, 7, 17, 18, 15, 16, 14, 13, 23, 24, 21, 22, 20, 19, 5, 6, 3, 4, 2, 1 ], [ 12, 11, 7, 8, 10, 9, 18, 17, 13, 14, 16, 15, 6, 5, 1, 2, 4, 3, 24, 23, 19, 20, 22, 21 ], [ 13, 14, 17, 18, 16, 15, 1, 2, 5, 6, 4, 3, 19, 20, 23, 24, 22, 21, 7, 8, 11, 12, 10, 9 ], [ 14, 13, 15, 16, 18, 17, 20, 19, 21, 22, 24, 23, 2, 1, 3, 4, 6, 5, 8, 7, 9, 10, 12, 11 ], [ 15, 16, 18, 17, 14, 13, 3, 4, 6, 5, 2, 1, 9, 10, 12, 11, 8, 7, 21, 22, 24, 23, 20, 19 ], [ 16, 15, 13, 14, 17, 18, 22, 21, 19, 20, 23, 24, 10, 9, 7, 8, 11, 12, 4, 3, 1, 2, 5, 6 ], [ 17, 18, 16, 15, 13, 14, 11, 12, 10, 9, 7, 8, 5, 6, 4, 3, 1, 2, 23, 24, 22, 21, 19, 20 ], [ 18, 17, 14, 13, 15, 16, 12, 11, 8, 7, 9, 10, 24, 23, 20, 19, 21, 22, 6, 5, 2, 1, 3, 4 ], [ 19, 20, 22, 21, 24, 23, 13, 14, 16, 15, 18, 17, 7, 8, 10, 9, 12, 11, 1, 2, 4, 3, 6, 5 ], [ 20, 19, 24, 23, 22, 21, 8, 7, 12, 11, 10, 9, 14, 13, 18, 17, 16, 15, 2, 1, 6, 5, 4, 3 ], [ 21, 22, 20, 19, 23, 24, 15, 16, 14, 13, 17, 18, 3, 4, 2, 1, 5, 6, 9, 10, 8, 7, 11, 12 ], [ 22, 21, 23, 24, 20, 19, 10, 9, 11, 12, 8, 7, 4, 3, 5, 6, 2, 1, 16, 15, 17, 18, 14, 13 ], [ 23, 24, 19, 20, 21, 22, 5, 6, 1, 2, 3, 4, 17, 18, 13, 14, 15, 16, 11, 12, 7, 8, 9, 10 ], [ 24, 23, 21, 22, 19, 20, 6, 5, 3, 4, 1, 2, 12, 11, 9, 10, 7, 8, 18, 17, 15, 16, 13, 14 ] ]
This loop has order 1053=34·13. Its Cayley table is too big to give it explicitely. One can construct it using the following GAP code:
LoadPackage("loops");
Gloop_section_constr:=function(g,h1)
local f,h2,sect,stab;
f:=DirectProduct(g,g);
h2:=Stabilizer(g,1);
sect:=List(g,x->Image(Embedding(f,1),x)*Image(Embedding(f,2),x^-1));
stab:=ClosureGroup(Image(Embedding(f,1),h1), Image(Embedding(f,2),h2));
sect:=Image(ActionHomomorphism(f,RightCosets(f,stab),OnRight),sect);
sect:=Set(List(sect,ListPerm));
sect[1]:=List([1..Size(g)],i->i);
return Set(sect);
end;
g_odd:=Group([ (2,18,25,9,22,8,17,11,12,26,14,10,4)(3,23,13,5,16,6,24,21,20,15,27,19,7),
(2,9,22)(3,5,16)(4,11,18)(6,15,24)(7,21,23)(8,26,17)(12,14,25)(13,20,27),
(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) ]);
h_odd:=Group([ (1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26),
(1,21,11)(2,19,12)(3,20,10)(4,24,14)(5,22,15)(6,23,13)(7,27,17)(8,25,18)(9,26,16),
(1,7,13,3,9,15,2,8,14)(4,19,26,6,21,25,5,20,27)(10,18,23,12,17,22,11,16,24) ]);
simple_bol_loop_of_order_1053:=LoopByCayleyTable(Gloop_section_constr(g_odd,h_odd));
This loop shows that the Odd Order Theorem does not hold for the class of finite Bol loops. Moreover, all three examples given here have order paqb for some primes p and q, thus, not even the paqb-theorem holds for Bol loops.
This loop was constructed using Michael Aschbacher's recipe from the paper On Bol loops of exponent 2. J. Algebra 288, No. 1, 99-136 (2005). The left multiplication group is a non-split extension of PGL(2,5) by the elementary Abelian group of order 32.
This loop has an elementary Abelian subgroup of order 16. On the remaining 80 elements, the automorphism group of the loop acts regularly.
Another interesting property of this loop is that all its nuclei are trivial.
The Cayley table of this loop:
[ [ 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, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96 ], [ 2, 1, 44, 27, 56, 55, 8, 7, 10, 9, 43, 28, 26, 46, 25, 45, 54, 53, 52, 51, 50, 49, 24, 23, 15, 13, 4, 12, 30, 29, 76, 75, 74, 73, 36, 35, 80, 79, 78, 77, 42, 41, 11, 3, 16, 14, 48, 47, 22, 21, 20, 19, 18, 17, 6, 5, 87, 85, 82, 84, 88, 86, 81, 83, 67, 68, 65, 66, 71, 72, 69, 70, 34, 33, 32, 31, 40, 39, 38, 37, 63, 59, 64, 60, 58, 62, 57, 61, 92, 91, 90, 89, 96, 95, 94, 93 ], [ 3, 14, 1, 23, 64, 82, 12, 15, 11, 16, 9, 7, 24, 2, 8, 10, 81, 60, 58, 88, 87, 62, 4, 13, 30, 42, 41, 29, 28, 25, 91, 89, 66, 67, 45, 43, 65, 68, 92, 90, 27, 26, 36, 47, 35, 48, 44, 46, 61, 85, 86, 57, 59, 83, 84, 63, 52, 19, 53, 18, 49, 22, 56, 5, 37, 33, 34, 38, 73, 79, 80, 74, 69, 72, 94, 96, 93, 95, 70, 71, 17, 6, 54, 55, 50, 51, 21, 20, 32, 40, 31, 39, 77, 75, 78, 76 ], [ 4, 13, 23, 1, 62, 87, 11, 16, 12, 15, 7, 9, 2, 24, 10, 8, 88, 58, 60, 81, 82, 64, 3, 14, 35, 48, 47, 36, 43, 45, 95, 93, 71, 70, 25, 28, 72, 69, 96, 94, 44, 46, 29, 41, 30, 42, 27, 26, 63, 84, 83, 59, 57, 86, 85, 61, 53, 18, 52, 19, 56, 5, 49, 22, 74, 80, 79, 73, 38, 34, 33, 37, 68, 65, 90, 92, 89, 91, 67, 66, 20, 21, 51, 50, 55, 54, 6, 17, 77, 75, 78, 76, 32, 40, 31, 39 ], [ 5, 17, 39, 76, 1, 9, 21, 19, 6, 18, 37, 74, 77, 32, 79, 34, 2, 10, 8, 24, 7, 23, 22, 20, 33, 31, 40, 38, 52, 50, 26, 14, 25, 16, 55, 53, 11, 28, 3, 27, 51, 49, 73, 75, 80, 78, 54, 56, 42, 30, 41, 29, 36, 47, 35, 48, 69, 67, 68, 70, 95, 92, 91, 96, 87, 84, 58, 59, 57, 60, 85, 82, 43, 12, 44, 4, 13, 46, 15, 45, 93, 72, 90, 66, 71, 94, 65, 89, 88, 83, 63, 62, 81, 86, 61, 64 ], [ 6, 55, 58, 60, 29, 1, 54, 17, 51, 20, 63, 61, 59, 57, 62, 64, 8, 48, 42, 10, 23, 36, 21, 50, 86, 82, 84, 88, 5, 56, 71, 70, 93, 95, 49, 22, 92, 90, 68, 65, 52, 19, 81, 85, 83, 87, 53, 18, 35, 24, 9, 41, 47, 7, 2, 30, 14, 3, 13, 4, 12, 15, 11, 16, 40, 78, 77, 39, 76, 32, 31, 75, 94, 96, 72, 69, 67, 66, 91, 89, 43, 26, 45, 27, 44, 25, 46, 28, 80, 38, 79, 37, 33, 73, 34, 74 ], [ 7, 8, 13, 14, 18, 17, 1, 2, 23, 24, 15, 16, 3, 4, 11, 12, 6, 5, 22, 21, 20, 19, 9, 10, 43, 44, 46, 45, 47, 48, 38, 37, 40, 39, 42, 41, 32, 31, 34, 33, 36, 35, 25, 26, 28, 27, 29, 30, 52, 51, 50, 49, 56, 55, 54, 53, 84, 82, 85, 87, 83, 81, 86, 88, 96, 95, 93, 94, 90, 89, 91, 92, 78, 77, 80, 79, 74, 73, 76, 75, 62, 58, 61, 57, 59, 63, 60, 64, 70, 69, 71, 72, 67, 68, 66, 65 ], [ 8, 7, 26, 46, 53, 54, 2, 1, 24, 23, 25, 45, 44, 27, 43, 28, 55, 56, 49, 50, 51, 52, 10, 9, 11, 3, 14, 16, 48, 47, 79, 80, 77, 78, 41, 42, 75, 76, 73, 74, 35, 36, 15, 13, 12, 4, 30, 29, 19, 20, 21, 22, 5, 6, 17, 18, 60, 59, 58, 57, 64, 63, 62, 61, 93, 94, 96, 95, 91, 92, 90, 89, 39, 40, 37, 38, 33, 34, 31, 32, 86, 85, 88, 87, 82, 81, 84, 83, 72, 71, 69, 70, 65, 66, 68, 67 ], [ 9, 10, 45, 25, 50, 49, 23, 24, 1, 2, 46, 26, 28, 43, 27, 44, 52, 51, 54, 53, 56, 55, 7, 8, 4, 12, 15, 13, 41, 42, 32, 31, 34, 33, 48, 47, 38, 37, 40, 39, 29, 30, 14, 16, 3, 11, 36, 35, 6, 5, 18, 17, 20, 19, 22, 21, 64, 63, 62, 61, 60, 59, 58, 57, 72, 71, 70, 69, 68, 67, 66, 65, 74, 73, 76, 75, 78, 77, 80, 79, 85, 86, 87, 88, 81, 82, 83, 84, 93, 94, 95, 96, 89, 90, 91, 92 ], [ 10, 9, 16, 15, 21, 22, 24, 23, 2, 1, 14, 13, 12, 11, 4, 3, 19, 20, 17, 18, 5, 6, 8, 7, 27, 28, 25, 26, 42, 41, 75, 76, 73, 74, 47, 48, 79, 80, 77, 78, 30, 29, 46, 45, 44, 43, 35, 36, 55, 56, 53, 54, 51, 52, 49, 50, 83, 81, 86, 88, 84, 82, 85, 87, 70, 69, 72, 71, 66, 65, 68, 67, 33, 34, 31, 32, 39, 40, 37, 38, 58, 62, 57, 61, 63, 59, 64, 60, 96, 95, 94, 93, 92, 91, 90, 89 ], [ 11, 28, 24, 35, 91, 90, 46, 13, 15, 45, 1, 29, 8, 48, 9, 41, 94, 95, 92, 89, 93, 96, 27, 3, 42, 47, 23, 2, 12, 43, 82, 58, 57, 84, 4, 44, 63, 86, 88, 64, 16, 25, 30, 36, 10, 7, 26, 14, 69, 71, 66, 68, 70, 72, 65, 67, 33, 32, 74, 75, 80, 77, 37, 40, 55, 51, 56, 52, 49, 53, 50, 54, 85, 59, 60, 87, 62, 81, 83, 61, 78, 31, 79, 34, 73, 38, 76, 39, 20, 6, 5, 19, 21, 17, 18, 22 ], [ 12, 43, 35, 24, 65, 67, 26, 14, 16, 25, 29, 1, 48, 8, 41, 9, 70, 72, 66, 68, 69, 71, 44, 4, 10, 7, 36, 30, 11, 28, 57, 84, 82, 58, 3, 27, 88, 64, 63, 86, 15, 45, 2, 23, 42, 47, 46, 13, 93, 96, 92, 89, 94, 95, 91, 90, 31, 34, 76, 73, 78, 79, 39, 38, 5, 19, 6, 20, 21, 17, 22, 18, 60, 87, 85, 59, 83, 61, 62, 81, 80, 33, 77, 32, 75, 40, 74, 37, 52, 56, 55, 51, 49, 53, 54, 50 ], [ 13, 46, 9, 41, 90, 91, 28, 11, 3, 27, 8, 48, 1, 29, 24, 35, 95, 94, 89, 92, 96, 93, 45, 15, 36, 30, 10, 7, 14, 26, 64, 88, 86, 63, 16, 25, 84, 57, 58, 82, 4, 44, 47, 42, 23, 2, 43, 12, 71, 69, 68, 66, 72, 70, 67, 65, 38, 39, 79, 78, 73, 76, 34, 31, 56, 52, 55, 51, 50, 54, 49, 53, 61, 83, 81, 62, 87, 60, 59, 85, 75, 40, 74, 37, 80, 33, 77, 32, 19, 5, 6, 20, 22, 18, 17, 21 ], [ 14, 26, 41, 9, 67, 65, 43, 12, 4, 44, 48, 8, 29, 1, 35, 24, 72, 70, 68, 66, 71, 69, 25, 16, 23, 2, 42, 47, 13, 46, 86, 63, 64, 88, 15, 45, 58, 82, 84, 57, 3, 27, 7, 10, 36, 30, 28, 11, 96, 93, 89, 92, 95, 94, 90, 91, 40, 37, 77, 80, 75, 74, 32, 33, 6, 20, 5, 19, 22, 18, 21, 17, 81, 62, 61, 83, 59, 85, 87, 60, 73, 38, 76, 39, 78, 31, 79, 34, 51, 55, 56, 52, 50, 54, 53, 49 ], [ 15, 12, 8, 10, 82, 64, 14, 3, 13, 4, 24, 2, 9, 7, 1, 23, 60, 81, 88, 58, 62, 87, 16, 11, 47, 36, 35, 48, 46, 44, 90, 92, 68, 65, 27, 26, 67, 66, 89, 91, 45, 43, 42, 30, 41, 29, 25, 28, 85, 61, 57, 86, 83, 59, 63, 84, 51, 20, 54, 17, 50, 21, 55, 6, 34, 38, 37, 33, 80, 74, 73, 79, 71, 70, 95, 93, 96, 94, 72, 69, 18, 5, 53, 56, 49, 52, 22, 19, 39, 31, 40, 32, 76, 78, 75, 77 ], [ 16, 11, 10, 8, 87, 62, 13, 4, 14, 3, 2, 24, 7, 9, 23, 1, 58, 88, 81, 60, 64, 82, 15, 12, 41, 29, 30, 42, 26, 27, 94, 96, 69, 72, 44, 46, 70, 71, 93, 95, 25, 28, 48, 35, 47, 36, 45, 43, 84, 63, 59, 83, 86, 57, 61, 85, 54, 17, 51, 20, 55, 6, 50, 21, 79, 73, 74, 80, 33, 37, 38, 34, 66, 67, 91, 89, 92, 90, 65, 68, 19, 22, 52, 49, 56, 53, 5, 18, 76, 78, 75, 77, 39, 31, 40, 32 ], [ 17, 5, 37, 74, 2, 10, 19, 21, 18, 6, 39, 76, 79, 34, 77, 32, 1, 9, 7, 23, 8, 24, 20, 22, 31, 33, 38, 40, 50, 52, 25, 16, 26, 14, 53, 55, 3, 27, 11, 28, 49, 51, 75, 73, 78, 80, 56, 54, 41, 29, 42, 30, 35, 48, 36, 47, 91, 96, 95, 92, 68, 70, 69, 67, 81, 86, 64, 61, 63, 62, 83, 88, 44, 4, 43, 12, 15, 45, 13, 46, 65, 89, 71, 94, 90, 66, 93, 72, 82, 85, 57, 60, 87, 84, 59, 58 ], [ 18, 53, 87, 82, 8, 48, 56, 5, 49, 22, 83, 86, 84, 85, 88, 81, 29, 1, 23, 36, 42, 10, 19, 52, 61, 60, 59, 62, 17, 54, 67, 66, 91, 89, 51, 20, 94, 96, 72, 69, 50, 21, 64, 57, 63, 58, 55, 6, 9, 41, 35, 24, 2, 30, 47, 7, 44, 46, 27, 26, 25, 28, 45, 43, 76, 32, 31, 75, 40, 78, 77, 39, 92, 90, 68, 65, 71, 70, 93, 95, 16, 4, 11, 13, 14, 12, 3, 15, 34, 74, 33, 73, 79, 37, 80, 38 ], [ 19, 52, 82, 87, 10, 42, 49, 22, 56, 5, 86, 83, 85, 84, 81, 88, 36, 23, 1, 29, 48, 8, 18, 53, 63, 58, 57, 64, 20, 51, 70, 71, 95, 93, 54, 17, 90, 92, 65, 68, 55, 6, 62, 59, 61, 60, 50, 21, 7, 47, 30, 2, 24, 35, 41, 9, 27, 26, 44, 46, 45, 43, 25, 28, 39, 77, 78, 40, 75, 31, 32, 76, 96, 94, 69, 72, 66, 67, 89, 91, 15, 3, 12, 14, 13, 11, 4, 16, 79, 37, 80, 38, 34, 74, 33, 73 ], [ 20, 22, 74, 37, 24, 8, 18, 6, 19, 21, 76, 39, 34, 79, 32, 77, 23, 7, 9, 1, 10, 2, 17, 5, 78, 80, 73, 75, 55, 53, 45, 15, 46, 13, 52, 50, 4, 44, 12, 43, 56, 54, 40, 38, 31, 33, 49, 51, 47, 36, 48, 35, 30, 42, 29, 41, 95, 92, 91, 96, 69, 67, 68, 70, 88, 83, 62, 63, 61, 64, 86, 81, 27, 3, 28, 11, 16, 25, 14, 26, 72, 93, 66, 90, 94, 71, 89, 65, 87, 84, 59, 58, 82, 85, 57, 60 ], [ 21, 50, 60, 58, 36, 23, 51, 20, 54, 17, 61, 63, 57, 59, 64, 62, 10, 42, 48, 8, 1, 29, 6, 55, 83, 87, 85, 81, 22, 49, 66, 67, 89, 91, 56, 5, 96, 94, 69, 72, 53, 18, 88, 84, 86, 82, 52, 19, 30, 2, 7, 47, 41, 9, 24, 35, 13, 4, 14, 3, 11, 16, 12, 15, 75, 31, 32, 76, 39, 77, 78, 40, 90, 92, 65, 68, 70, 71, 95, 93, 28, 46, 25, 44, 27, 45, 26, 43, 33, 73, 34, 74, 80, 38, 79, 37 ], [ 22, 20, 76, 39, 23, 7, 6, 18, 21, 19, 74, 37, 32, 77, 34, 79, 24, 8, 10, 2, 9, 1, 5, 17, 80, 78, 75, 73, 53, 55, 46, 13, 45, 15, 50, 52, 12, 43, 4, 44, 54, 56, 38, 40, 33, 31, 51, 49, 48, 35, 47, 36, 29, 41, 30, 42, 68, 70, 69, 67, 91, 96, 95, 92, 82, 85, 60, 57, 59, 58, 84, 87, 28, 11, 27, 3, 14, 26, 16, 25, 89, 65, 94, 71, 66, 90, 72, 93, 81, 86, 61, 64, 88, 83, 63, 62 ], [ 23, 24, 28, 43, 51, 52, 9, 10, 7, 8, 27, 44, 45, 25, 46, 26, 49, 50, 55, 56, 53, 54, 1, 2, 14, 16, 11, 3, 36, 35, 37, 38, 39, 40, 30, 29, 31, 32, 33, 34, 47, 48, 4, 12, 13, 15, 41, 42, 17, 18, 5, 6, 21, 22, 19, 20, 88, 86, 81, 83, 87, 85, 82, 84, 92, 91, 89, 90, 94, 93, 95, 96, 77, 78, 79, 80, 73, 74, 75, 76, 59, 63, 60, 64, 62, 58, 61, 57, 67, 68, 66, 65, 70, 69, 71, 72 ], [ 24, 23, 12, 11, 20, 19, 10, 9, 8, 7, 4, 3, 16, 15, 14, 13, 22, 21, 6, 5, 18, 17, 2, 1, 46, 45, 43, 44, 35, 36, 80, 79, 78, 77, 29, 30, 76, 75, 74, 73, 48, 47, 27, 28, 26, 25, 42, 41, 54, 53, 56, 55, 50, 49, 52, 51, 61, 62, 63, 64, 57, 58, 59, 60, 89, 90, 92, 91, 95, 96, 94, 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