OFFSET
2,4
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.
LINKS
W. E. Clark, M. Elhamdadi, M. Saito, 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
FORMULA
a(p) = A181771(p) = p - 2, for prime p > 2.
EXAMPLE
a(2) = 1 since the quandle of order 2 is trivially simple (though not connected).
PROG
(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)<Size(gg) then return false; fi;
od;
return true;
end;;
a:=[1, 1];;
for n in [3..35] do
a[n]:=0;
for i in [1..NrSmallQuandles(n)] do
if IsSimpleQuandle(SmallQuandle(n, i)) then
a[n]:=a[n]+1;
fi;
od;
od;
List([1..35], u->a[u]); # W. Edwin Clark, Dec 06 2011
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
James McCarron, Oct 27 2011
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
STATUS
approved