A281319 Number of left Bol loops (including Moufang loops) of order n which are not groups.

0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 3, 0, 0, 2, 2038, 0, 2, 0, 3, 2, 0, 0

Offset

1,8

Comments

A loop is a set L with binary operation (denoted simply by juxtaposition) such that for each a in L, the left (right) multiplication map L_a:=L->L, x->xa (R_a: L->L, x->ax) is bijective and L has a two-sided identity 1. 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.

References

E. G. Goodaire and S. May, Bol loops of order less than 32, Dept of Math and Statistics, Memorial University of Newfoundland, Canada, 1995.

Links

Table of n, a(n) for n=1..23.

R. P. Burn, Corrigenda: Finite Bol loops: III, Math. Proc. Camb. Phil. Soc. (1985), 98, 381.

Michael K. Kinyon, Gábor P. Nagy and Petr Vojtěchovský, Bol loops and Bruck loops of order pq, Journal of Algebra, Volume 473, 2017, Pages 481-512.

Eric Moorhouse, Bol loops of small orders

B. L. Sharma, Classification of Bol loops of order 18, Acta Universitatis Carolinae. Mathematica et Physica 025.1 (1984): 37-44.

B. L. Sharma and A. R. T. Solarin, On classification of Bol loops of order 3p (p>3), Comm. in Algebra 16:1(1988), 37-55.

Example

a(8)=6 since there are 6 left Bol loops of order 8 and a(12)=3 since there are 3 left Bol loops of order 12 one of which is the smallest Moufang loop.

Crossrefs

Cf. A090750.

Sequence in context: A097606 A201995 A226784 * A174431 A260712 A173066

Adjacent sequences: A281316 A281317 A281318 * A281320 A281321 A281322

Keyword

nonn,more

Author

Muniru A Asiru, Jan 20 2017

Extensions

a(18) changed to 2 by N. J. A. Sloane, Feb 02 2023 at the suggestion of Kurosh Mavaddat Nezhaad, who said in an email that the number of Bol loops of order 18, and generally of order 2p^2 up to isomorphism, is exactly 2. See Sharma (1984) or Burn (1985).

Status

approved