|
| |
|
|
A179010
|
|
The number of isomorphism classes of commutative quandles of order n.
|
|
2
|
|
|
|
1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 7
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,9
|
|
|
COMMENTS
|
A quandle (X,*) is commutative if a*b = b*a for all a,b in X. Every finite commutative quandle (X,*) is obtained from an odd order, commutative Moufang loop (X,+) where x*y = (1/2)(x+y). Thus a(n) is the number of isomorphism classes of commutative Moufang loops of order n if n is odd and is 0 if n is even. Commutative Moufang loops of order less than 81 are associative hence abelian groups. But, there are two non-associative commutative Moufang loops of order 81. Thus a(n) = number of isomorphism classes of abelian groups of odd order for n < 81 and a(81) = A000688(81) + 2 = 7. For proofs of these facts see, e.g., the papers below by Belousov, Nagy and Vojtchovský, and Glauberman.
|
|
|
LINKS
|
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,hard,more
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
Results due to Belousov, Nagy and Vojtchovský, and Glauberman added, and sequence extended to n = 81, by W. Edwin Clark, Jan 25 2011
In Comments section, "Every commutative quandle" replaced with "Every finite commutative quandle" by W. Edwin Clark, Mar 09 2014
|
|
|
STATUS
|
approved
|
| |
|
|
