A090750 -id:A090750 - cp's OEIS frontend

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

Muniru A Asiru, Jan 20 2017

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.

			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.

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

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.

Cf. A090750.

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).

A281462 Number of code loops of order n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 80, 0

Offset: 1

Muniru A Asiru, Jan 22 2017

nonn

A code loop is a Moufang 2-loop Q with a central subloop Z of order 2 such that Q/Z is an elementary abelian group. The library named code in LOOPS version 2.2.0, Computing with quasigroups and loops in GAP (Groups, Algorithm and Programming), contains all nonassociative code loops of order less than 65. Every code loop is a Moufang loop but not conversely. The GAP command IsCodeLoop(MoufangLoop(n,m)) gives the m-th nonassociative code loop of order n in the LOOPS Package library. Code loops of small orders were classified by G. P. Nagy and P. Vojtechovsky.

(Groups are specifically excluded from the counts.)

			a(16)=5 because all the 5 Moufang loops of order 16 are code loops;
a(32)=16 because only 16 of the 71 Moufang loops of order 32 are code loops.

R. L. Griess Jr., Code loops, J. Algebra 100(1986), 224-234.
G. P. Nagy and P. Vojtechovsky, The Moufang loops of order 64 and 81, Symbolic Comput., 42(2007), 871-883.
G. P. Nagy and P. Vojtechovsky, Loops version 2.2.0, Computing with quasigroups and loops in GAP, 2012.

Cf. A090750, A281319.

A132126 Number of nonassociative subloops of order 8n of the Cayley octonions (up to isomorphism).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Paul Boddington, Oct 31 2007

Every nonassociative subloop of the octonions has order a multiple of 8.

Antti Karttunen, Table of n, a(n) for n = 1..65537
P. Boddington and D. Rumynin, On Curtis' theorem about finite octonionic loops, Proc. Amer. Math. Soc. 135 (2007), 1651-1657.

Cf. A090750.

ReplacePart[PadRight[{0},120,1],{6->2,12->2,30->3}] (* Harvey P. Dale, Dec 18 2018 *)

A132126(n) = if(1==n,0,if((6==n)||(12==n),2,if(30==n,3,1))); \\ Antti Karttunen, Sep 27 2018

a(1) = 0, a(6) = 2, a(12) = 2, a(30) = 3, otherwise a(n) = 1.

A152123 3^a(n) is the order of the free commutative Moufang loop of exponent 3 on n generators.

Original entry on oeis.org

4, 12, 49, 220, 1014

Offset: 3

Jonathan Vos Post, Nov 25 2008

Yu. I. Manin, Cubic forms, Amsterdam: North-Holland, 1979.
Yu. I. Manin, Cubic Forms, Second edition, North-Holland Publishing Co., Amsterdam, 1986, page 312. MR0833513 (87d:11037)

Alexander N. Grishkov and Ivan P. Shestakov, Commutative Moufang loops and alternative algebras, arXiv:0811.3787 [math.RA], 2008.
Marina Rasskazova, Alexandre Grichkov and Henrique Guzzo, Commutative Moufang loops and cubic hypersurface, 32o Colóquio Brasileiro de Matemática, IMPA, 2019.
Jonathan D. H. Smith, Commutative Moufang loops and Bessel functions, Invent. Math. 67 (1982), no. 1, 173-187.

Cf. A000373, A090750, A118641.

Definition changed and a(7) changed by Andrey Zabolotskiy, Sep 28 2021

A281554 Number of nonassociative right conjugacy closed loops of order n up to isomorphism.

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 0, 19, 5, 16, 0, 155, 0, 97, 17, 6317, 0, 1901, 0, 8248, 119, 10487, 0, 471995, 119, 151971, 152701

Offset: 1

Muniru A Asiru, Jan 24 2017

For a groupoid Q and x in Q, define the right (left) translation map R_x: Q->Q by yR_x=yx (L_x: Q->Q by yL_x=xy). A loop is a groupoid Q with neutral element 1 in which all translations are bijections in Q. A loop Q is right conjugacy closed if (R_x)^(-1)R_yR_x is a right translation for every x, y in Q. Since any finite loop of order n < 5 is a group, then nonassociative right conjugacy closed loops exist when the order n > 5. In the literature, every nonassociative right conjugacy closed loop of order n can be represented as a union of certain conjugacy classes of a transitive group of degree n. The number of nonassociative right conjugacy closed loops of order n up to isomorphism were summarized in LOOPS version 3.3.0, Computing with quasigroups and loops in GAP (Groups, Algorithm and Programming).

			a(6)=3 because there are 3 nonassociative right conjugacy closed loops of order 6 and a(8)=19 because there are 19 nonassociative right conjugacy closed loops of order 8.

G. P. Nagy and P. Vojtechovsky, Loops version 3.3.0, Computing with quasigroups and loops in GAP, 2016.

Cf. A090750, A281319, A281462.

A282530 Number of finite FRUTE loops of order n up to isomorphism.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 27, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Muniru A Asiru, Feb 17 2017

For a groupoid Q and x in Q, define the right (left) translation map R_x: Q->Q by yR_x=yx (L_x: Q->Q by yL_x=xy). A loop is a groupoid Q with neutral element 1 in which all translations are bijections in Q. A loop Q is called a FRUTE loop if it satisfies the identity (x.xy)z=(y.xz)x for all x, y, z in Q. The smallest associative non-commutative finite FRUTE loop is of order 8, the quaternion group having 8 elements.

			a(8)=2 since there are 2 FRUTE loops of order 8, one of which is the quaternion group of order 8 and a(16)=6 since there are 6 FRUTE loops of order 16.

T. G. Jaiyeola, A. A. Adeniregun and M. A. Asiru, Finite FRUTE loops, Journal of Algebra and its Applications, 16:2(2017), 10 pages.

Cf. A090750, A281319, A281462, A281554

cp's OEIS Frontend

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

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A281462 Number of code loops of order n.

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A132126 Number of nonassociative subloops of order 8n of the Cayley octonions (up to isomorphism).

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A152123 3^a(n) is the order of the free commutative Moufang loop of exponent 3 on n generators.

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A281554 Number of nonassociative right conjugacy closed loops of order n up to isomorphism.

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A282530 Number of finite FRUTE loops of order n up to isomorphism.

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