

A118641


Number of nonisomorphic finite nonassociative, invertible loops of order n.


2




OFFSET

5,2


COMMENTS

These are nonassociative loops in which every element has a unique inverse and it includes IP, Moufang and Bol loops [Cawagas]. The data were generated and checked by a supercomputer of 48 Pentium II 400 processors, specially built for automated reasoning, in about three days. In general, a loop is a quasigroup with an identity element e such that xe = x and ex = x for any x in the quasigroup. All groups are loops. A quasigroup is a groupoid G such that for all a and b in G, there exist unique c and d in G such that ac = b and da = b. Hence a quasigroup is not required to have an identity element, nor be associative. Equivalently, one can state that quasigroups are precisely groupoids whose multiplication tables are Latin squares (possibly empty).


REFERENCES

Cawagas, R. E., Terminal Report: Development of the Theory of Finite Pseudogroups (1998). Research supported by the National Research Council of the Philippines (19961998) under Project B88 and B95.


LINKS



EXAMPLE

a(5) = 1 (which is nonAbelian).
a(6) = 33 (7 Abelian + 26 nonAbelian).
a(7) = 2333 (16 Abelian + 2317 nonAbelian).


CROSSREFS



KEYWORD

nonn,bref,more


AUTHOR



STATUS

approved



