A260645 The number of central quasigroups (also known as T-quasigroups, or quasigroups affine over an abelian group) of order n, up to isomorphism.

1, 1, 5, 19, 19, 5, 41, 385, 231, 19, 109, 95, 155, 41, 95, 41387, 271, 231, 341, 361, 205, 109, 505, 1925, 3337, 155, 36118, 779, 811, 95, 929, 19823665, 545, 271, 779, 4389, 1331, 341, 775, 7315, 1639, 205, 1805, 2071, 4389, 505, 2161, 206935, 18099, 3337, 1355, 2945, 2755, 36118, 2071, 15785, 1705, 811, 3421, 1805, 3659, 929, 9471

Offset

1,3

Comments

A quasigroup (G,*) is called central if it admits an affine representation over an abelian group (G,+), that is, if x*y = f(x)+g(y)+c where f,g are automorphisms of (G,+) and c in G.

Links

David Stanovsky, Table of n, a(n) for n = 1..63

David Stanovský and Petr Vojtechovský, Central and medial quasigroups of small order, arxiv preprint arXiv:1511.03534 [math.GR], 2015.

Prog

(GAP) # gives the number of central quasigroups over SmallGroup(n, k)
LoadPackage("loops");
CQ := function( n, k )
    local G, ct, elms, inv, A, f_reps, count, f, Cf, O, g_reps, g, Cfg, W, unused, c, Wc;
    G := SmallGroup( n, k );
    G := IntoLoop( G );
    ct := CayleyTable( G );
    elms := Elements( G );
    inv := List( List( [1..n], i -> elms[i]^(-1) ), x -> x![1] );
    A := AutomorphismGroup( G );
    f_reps := List( ConjugacyClasses( A ), Representative );
    count := 0;
    for f in f_reps do
        Cf := Centralizer( A, f );
        O := OrbitsDomain( Cf, A );
        g_reps := List( O, x -> x[1] );
        for g in g_reps do
            Cfg := Intersection( Cf, Centralizer( A, g ) );
            W := Set( [1..n], w -> ct[w][ inv[ ct[w^f][w^g] ] ] );
            unused := [1..n];
            while not IsEmpty( unused ) do
                c := unused[1];
                count := count + 1;
                if Size(W) = Length(unused) then
                    unused := [];
                else
                    Wc := Set( W, w -> ct[w][c] );
                    Wc := Union( Orbits( Cfg, Wc ) );
                    unused := Difference( unused, Wc );
                fi;
            od;
        od;
    od;
    return count;
end;

Crossrefs

Cf. A226193.

Sequence in context: A370824 A169698 A211177 * A226658 A212153 A337401

Adjacent sequences: A260642 A260643 A260644 * A260646 A260647 A260648

Keyword

nonn,hard,mult

Author

David Stanovsky, Nov 12 2015

Status

approved