cp's OEIS Frontend

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.)

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).

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.