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

1,5

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

Table of n, a(n) for n=1..100.

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

Prog

(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;

Crossrefs

Cf. A193067, A181771.

See also Index to OEIS under quandles.

Sequence in context: A225058 A002123 A276408 * A275393 A029840 A144670

Adjacent sequences: A225741 A225742 A225743 * A225745 A225746 A225747

Keyword

nonn

Author

W. Edwin Clark, Aug 04 2013

Status

approved