A196111 Number of isomorphism classes of simple quandles of order n.

1, 1, 1, 3, 0, 5, 2, 3, 1, 9, 1, 11, 0, 2, 3, 15, 0, 17, 2, 2, 0, 21, 1, 10, 0, 8, 2, 27, 1, 29, 6, 0, 0, 0, 3, 35, 0, 0, 2, 39, 3, 41, 0, 3, 0, 45

Offset: 2

Author

James McCarron, Oct 27 2011

Keywords

• nonn
• hard
• more

Comments

A quandle is simple if it has more than one element, and if it has no homomorphic images other than itself or the singleton quandle. Since a simple quandle with more than two elements is connected, we have a(n) <= A181771(n), for n > 2, with equality if n is prime.

Some authors consider the quandle with one element to be simple and some do not.

Examples

			a(2) = 1 since the quandle of order 2 is trivially simple (though not connected).
		

Links

• W. E. Clark, M. Elhamdadi, M. Saito, and T. Yeatman, Quandle Colorings of Knots and Applications, arXiv preprint arXiv:1312.3307, 2013
• David Joyce, Simple Quandles, J. Algebra 79(2) 1982, 307-318.
• Leandro Vendramin, On the classification of quandles of low order, arXiv:1105.5341v1 [math.GT].
• Leandro Vendramin and Matías Graña, Rig, a GAP package for racks and quandles.
• Wikipedia, Racks and quandles

Crossrefs

Cf. A181769, A181771.

See also Index to OEIS under quandles.

Programs

GAP

# Using the Rig package.
LoadPackage("rig");
IsSimpleQuandle:=function(q)
local g,N,gg,n;
if IsFaithful(q) = false then return false; fi;
g:=InnerGroup(q);;
if Size(Center(g))>1 then return false; fi;
N:=NormalSubgroups(g);;
gg:=DerivedSubgroup(g);;
for n in N do
  if Size(n) = 1 then continue; fi;
  if IsSubset(gg,n) and Size(n)a[u]); # W. Edwin Clark, Dec 06 2011

Formula

a(p) = A181771(p) = p - 2, for prime p > 2.

Extensions

a(21) corrected by W. Edwin Clark, Dec 06 2011

a(32)-a(35) added by W. Edwin Clark, Dec 06 2011

a(36)-a(47) added by W. Edwin Clark, Dec 28 2014