A225744 - OEIS

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

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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;

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;;

CROSSREFS

Cf. A193067, A181771.