cp's OEIS Frontend

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

%I A225744 #16 Jan 31 2014 09:59:55
%S A225744 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,
%T A225744 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,
%U A225744 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
%N A225744 The number of isomorphism classes of connected, Generalized Alexander quandles of order n.
%C A225744 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.
%H A225744 J. Scott Carter, <a href="http://arxiv.org/abs/1002.4429">A Survey of Quandle Ideas</a>, arXiv:1002.4429 [math.GT]
%H A225744 W. E. Clark, M. Elhamdadi, M. Saito, T. Yeatman, <a href="http://arxiv.org/abs/1312.3307">Quandle Colorings of Knots and Applications</a>, arXiv preprint arXiv:1312.3307, 2013
%o A225744 (GAP)
%o A225744 IsConnected:=function(A)
%o A225744 local B,LL;
%o A225744 B:=TransposedMat(A);
%o A225744    LL:=List(B,x->PermList(x));
%o A225744    return IsTransitive(Group(LL),[1..Length(A)]);
%o A225744 end;;
%o A225744 MakeGAlex:=function(f,g)
%o A225744 local e,n,QM,i,j;
%o A225744   e:=Elements(g);
%o A225744   n:=Length(e);
%o A225744   QM:=List([1..n],t->[1..n]);
%o A225744     for i in [1..n] do
%o A225744       for j in [1..n] do
%o A225744        QM[i][j]:=Position(e,Image(f,e[i]*e[j]^(-1))*e[j]);
%o A225744       od;
%o A225744     od;
%o A225744   return QM;
%o A225744 end;;
%o A225744 a:=[];;
%o A225744 for n in [1..100] do
%o A225744 a[n]:=0;
%o A225744 N:=NrSmallGroups(n);
%o A225744 for u in [1..N] do
%o A225744    g:=SmallGroup(n,u);
%o A225744    ag:=AutomorphismGroup(g);;
%o A225744    eag:=List(ConjugacyClasses(ag),Representative);
%o A225744    for t in eag do
%o A225744       QM:=MakeGAlex(t,g);
%o A225744       if IsConnected(QM) then a[n]:=a[n]+1; fi;
%o A225744    od;
%o A225744   od;
%o A225744 od;;
%o A225744 a;
%Y A225744 Cf. A193067, A181771.
%Y A225744 See also Index to OEIS under quandles.
%K A225744 nonn
%O A225744 1,5
%A A225744 _W. Edwin Clark_, Aug 04 2013