) from you if you have any comments regarding this work.
Update Thurs 23 Mar 2007: Just yesterday, Gábor Nagy announced the first known (non-Moufang) simple left Bol loop! Here is a copy of his announcement to the loop forum. In order to accomodate his example in my tables below, I have rewritten it as right Bol loop 24.9.1.3. I now regret that exactly three years ago I didn't extend my work a little further to the case of 9 involutions (see the following note)!
Update Tues 23 Mar 2004: I have found 7 new (non-associative, non-Moufang) Bol loops of order 24 with trivial centre. The newest seven examples are those contained in the last two isotopy classes listed below (isotopy classes 47 and 48). None of these are simple (each has a subloop of order 12). If L is a Bol loop of order 24 which is not in my list, then L has trivial centre, and every isotope of L has at most 9 involutions.
Update Mon 29 Mar 2004: I have checked that more generally, any time the right-multiplication maps Ra, for a in I(L), generate a transitive subgroup of the right-multiplication group of L, where L is a Bol loop of order 24, then L is one of the loops in my list below.
I've been working on a classification of Bol loops of small order.
Here is a list of the 103 known Bol loops of order 24, of which 83 are non-Moufang
(and non-associative). This list includes all 79 Bol loops of order 24 with non-trivial centre
(61 non-Moufang, 14 non-associative Moufang and 4 associative).
It also includes the 25 known Bol loops of order 24 with trivial centre
(23 non-Moufang, 1 non-associative Moufang and 1 associative).
I'll post an update however when and if I have finished
classifying all Bol loops of order 24.
I would appreciate an email
message () from you if you have any comments regarding this work.
Our examples are drawn from the following sources:
I have made available a quick review of the relevant definitions; for further explanation, proofs, etc., please consult a suitable list of references. I have also provided a cross-reference list comparing this list with previously published lists of groups and loops.
There are (to within isomorphism) exactly 78 (right) Bol loops of order 24 with nontrivial centre, including 14 groups, 4 nonassociative Moufang loops and 60 non-Moufang loops. There are 40 isotopy classes of such loops. I have made available descriptions of these loops in html format, found through the links below. Here also are the Cayley tables (in plain text form, 163 KB; or gzipped plain text form, 31 KB).
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 occurs, for example, in the commutator subloop of loop 24.19.2.3.
| |C(L)|=1 (11 loops) | |C(L)|=2 (49 loops) | |C(L)|=3 (1 loop) | |C(L)|=4 (10 loops) | |C(L)|=5 (2 loops) | |C(L)|=6 (6 loops) | |C(L)|=7 (1 loop) | |C(L)|=12 (3 loops) | |
|---|---|---|---|---|---|---|---|---|
| |I(L)|=1 (10 loops) | 24.1.2.0, 1, 2, 3, 4, 6, 8 | 24.1.4.0, 1 | 24.1.12.0 | |||||
| |I(L)|=3 (9 loops) | 24.3.2.0, 1, 2, 3, 4, 5 | 24.3.6.0, 1, 2 | ||||||
| |I(L)|=5 (9 loops) | 24.5.2.0, 1, 2, 3, 4, 5 | 24.5.4.0 | 24.5.12.0, 1 | |||||
| |I(L)|=7 (16 loops) | 24.7.1.0 | 24.7.2.0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 12 | 24.7.4.1, 2 | 24.7.6.0, 1 | ||||
| |I(L)|=9 (11 loops) | 24.9.1.0, 2, 3 | 24.9.2.0, 1, 2, 4 | 24.9.4.0, 1, 2 | 24.9.7.0 | ||||
| |I(L)|=11 (5 loops) | 24.11.2.0, 1, 2, 3 | 24.11.6.0 | ||||||
| |I(L)|=13 (11 loops) | 24.13.1.0, 1 | 24.13.2.0, 1, 2, 3, 5, 7 | 24.13.4.0 | 24.13.5.0, 1 | ||||
| |I(L)|=15 (4 loops) | 24.15.1.1 | 24.15.2.0, 1 | 24.15.4.1 | |||||
| |I(L)|=17 (3 loops) | 24.17.1.0, 1 | 24.17.3.0 | ||||||
| |I(L)|=19 (3 loops) | 24.19.2.0, 1, 3 | |||||||
| |I(L)|=21 (2 loops) | 24.21.1.0, 1 |
| |C(L)|=2 (48 loops) | |C(L)|=4 (7 loops) | |C(L)|=6 (6 loops) | |C(L)|=12 (3 loops) | |
|---|---|---|---|---|
| |I(L)|=1 (11 loops) | 24.1.2.0, 1, 2, 3, 4, 6, 8, 9 | 24.1.4.0, 1 | 24.1.12.0 | |
| |I(L)|=3 (9 loops) | 24.3.2.0, 1, 2, 3, 4, 5 | 24.3.6.0, 1, 2 | ||
| |I(L)|=5 (6 loops) | 24.5.2.0, 1, 2 | 24.5.4.0 | 24.5.12.0, 1 | |
| |I(L)|=7 (14 loops) | 24.7.2.0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12 | 24.7.6.0, 1 | ||
| |I(L)|=9 (7 loops) | 24.9.2.0, 1, 2, 4 | 24.9.4.0, 1, 2 | ||
| |I(L)|=11 (3 loops) | 24.11.2.0, 1 | 24.11.6.0 | ||
| |I(L)|=13 (8 loops) | 24.13.2.0, 1, 2, 3, 5, 6, 7 | 24.13.4.0 | ||
| |I(L)|=15 (2 loops) | 24.15.2.0, 1 | |||
| |I(L)|=19 (4 loops) | 24.19.2.0, 1, 2, 3 |
The 50 Isotopy Classes of Known Bol Loops
Now let L be any Bol loop of order 24 (not necessarily having trivial centre), and suppose |I(L)| ≥ 11 where I(L) denotes the set of involutions in L. Denote by H the subgroup of the right-multiplication group of L generated by all the right-multiplication maps Rs for s ∈ I(L). Then {Rs : s ∈ I(L)} is a normal subset of H (i.e. a union of certain of the conjugacy classes of H). Moreover H permutes L transitively. This is obvious for |I(L)| ≥ 13, and it follows in the case |I(L)| = 11 since otherwise I(L), together with the identity of L, would form a subloop of order 12; this cannot occur since no Bol loop of order 12 has exponent 2. The computational algebra package GAP (Graphs, Algorithms and Programming) has a built-in library of transitive permutation groups of degree 24, and using this I have listed all possibilities for a transitive permutation group of degree 24 generated by a normal set of fixed-point-free involutions having size in the interval {13,14,...,23}. I then exhaustively searched for all possible ways to extend each of these sets of involutions to a Bol loop of order 24.
I have also used GAP to compute orders of left, right and full multiplication groups.
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revised March, 2007