Hi,
Right (and left) Bol loops are contained in the libraries of the LOOPS
package:
gap> LoadPackage("loops");
─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
Loading loops 3.4.1 (The LOOPS Package: Loops and quasigroups for GAP)
by Gábor Nagy (http://www.math.u-szeged.hu/~nagyg/) and
Petr Vojtěchovský (http://www.math.du.edu/~petr/).
Homepage: https://gap-packages.github.io/loops/
Report issues at https://github.com/gap-packages/loops/issues
─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
true
gap> DisplayLibraryInfo("right Bol");
The library contains all nonassociative right Bol loops of order less
than 17
and all nonassociative right Bol loops of order p*q, where p>q>2 are primes.
------
Extent of the library:
6 loops of order 8
3 loops of order 12
2038 loops of order 16
(p-q)/2 loops of order p*q for primes p>q>2 such that q divides p-1
(p-q+2)/2 loops of order p*q for primes p>q>2 such that q divides p+1
true
The constructions are rather ad hoc. See
R.P. Burn. Finite Bol loops I. Math. Proc. Cambridge Philos. Soc.,
84(3):53–66, 1978.
G.P. Nagy, Burnside problems for Moufang and Bol loops of small
exponent, Acta Sci. Math. (Szeged). 67 (2001) 687–696.
H. Kiechle, G.P. Nagy, On the extension of involutorial Bol loops, Abh.
Math. Sem. Univ. Hamburg. 72 (2002) 235–250.
https://doi.org/10.1007/BF02941674.
G.P. Nagy, On the structure and number of small Frattini Bol 2-loops,
Math. Proc. Cambridge Philos. Soc. 141 (2006) 409–419.
https://doi.org/10.1017/S0305004106009625.
G.P. Nagy, A class of simple proper Bol loops, Manuscripta Math. 127
(2008) 81–88. https://doi.org/10.1007/s00229-008-0188-5.
G.P. Nagy, A class of finite simple Bol loops of exponent 2, Trans.
Amer. Math. Soc. 361 (2009) 5331–5343.
https://doi.org/10.1090/S0002-9947-09-04646-7.
If you have some specific interest, feel free to contact me directly.
Best regards,
Gabor
On 2022. 06. 20. 16:13, Kurosh Mavaddat Nezhaad wrote:
Dear All,
I am working on loop theory, I would like to know the algorithm(s) that GAP
used for constructing Bol loops, any address of papers and literature would
be highly appreciated.
With kind regards,
Kuroš Mavaddat Nežâd
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