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

Author

Muniru A Asiru, Jan 20 2017

Keywords

• nonn
• more

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.

Examples

			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.
		

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

• 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.

Crossrefs

Cf. A090750.

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).