A225744 The number of isomorphism classes of connected, Generalized Alexander quandles of order n.

1, 0, 1, 1, 3, 0, 5, 3, 8, 0, 9, 3, 11, 0, 3, 9, 15, 0, 17, 3, 5, 0, 21, 5, 34, 0, 35, 5, 27, 0, 29, 17, 9, 0, 15, 18, 35, 0, 11, 9, 39, 0, 41, 9, 24, 0, 45, 21, 76, 0, 15, 11, 51, 0, 27, 19, 17, 0, 57, 15, 59, 0, 40, 97, 33, 0, 65, 15, 21, 0, 69, 37, 71, 0, 39, 17, 45, 0, 77, 34, 218, 0, 81, 15, 45, 0, 27, 27, 87, 0, 55, 21, 29, 0, 51, 43, 95, 0, 72, 34

Offset: 1

Author

W. Edwin Clark, Aug 04 2013

Keywords

• nonn

Comments

Given a group G and an automorphism f of G define the binary operation * on G by x*y = f(xy^(-1))y. Then (G,*) is a quandle. We call this a Generalized Alexander quandle. If G is abelian then (G,*) is an Alexander quandle (see A193024). (G,*) is connected if the group generated by the right translations of (G,*) is transitive on G.

Links

• J. Scott Carter, A Survey of Quandle Ideas, arXiv:1002.4429 [math.GT]
• W. E. Clark, M. Elhamdadi, M. Saito, T. Yeatman, Quandle Colorings of Knots and Applications, arXiv preprint arXiv:1312.3307, 2013

Crossrefs

Cf. A193067, A181771.

See also Index to OEIS under quandles.

Programs

GAP

IsConnected:=function(A)
local B,LL;
B:=TransposedMat(A);
   LL:=List(B,x->PermList(x));
   return IsTransitive(Group(LL),[1..Length(A)]);
end;;
MakeGAlex:=function(f,g)
local e,n,QM,i,j;
  e:=Elements(g);
  n:=Length(e);
  QM:=List([1..n],t->[1..n]);
    for i in [1..n] do
      for j in [1..n] do
       QM[i][j]:=Position(e,Image(f,e[i]*e[j]^(-1))*e[j]);
      od;
    od;
  return QM;
end;;
a:=[];;
for n in [1..100] do
a[n]:=0;
N:=NrSmallGroups(n);
for u in [1..N] do
   g:=SmallGroup(n,u);
   ag:=AutomorphismGroup(g);;
   eag:=List(ConjugacyClasses(ag),Representative);
   for t in eag do
      QM:=MakeGAlex(t,g);
      if IsConnected(QM) then a[n]:=a[n]+1; fi;
   od;
  od;
od;;
a;