A281462 Number of code loops of order n.

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

Author

Muniru A Asiru, Jan 22 2017

Keywords

• nonn

Comments

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.)

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A090750, A281319.