A226193 The number of medial quasigroups of order n, up to isomorphism.

1, 1, 5, 13, 19, 5, 41, 73, 116, 19, 109, 65, 155, 41, 95, 669, 271, 116, 341, 247, 205, 109, 505, 365, 1084, 155, 1574, 533, 811, 95, 929, 4193, 545, 271, 779, 1508, 1331, 341, 775, 1387, 1639, 205, 1805, 1417, 2204, 505, 2161, 3345, 4388, 1084, 1355, 2015, 2755, 1574, 2071, 2993, 1705, 811, 3421, 1235, 3659, 929, 4756

Offset: 1

Author

W. Edwin Clark, May 30 2013

Keywords

• nonn
• hard
• mult

Comments

See the Wikipedia link for "Medial magma" for definitions. This article also contains the Bruck-Toyoda theorem which characterizes medial quasigroups in terms of abelian groups.

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.
• Wikipedia, Medial magma

Programs

GAP

# gives the number of medial quasigroups over SmallGroup(n,k)
LoadPackage("loops");
MQ := 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];
                if f*g=g*f then count := count + 1; fi;
                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;
# David Stanovsky, Nov 12 2015

Maple

a:=proc(n)
if n = 1 then
     return 1;
  else
return MAGMA:-Enumerate(n,'medial','quasigroup');
end if;
end proc;

Extensions

a(9)-a(63) from David Stanovsky, Nov 12 2015