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

Author

David Stanovsky, Nov 12 2015

Keywords

• nonn
• hard
• mult

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.

Crossrefs

Cf. A226193.

Programs

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;