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

1,3

Comments

See the Wikipedia link for "Medial" 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

Maple

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

Prog

(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

Crossrefs

Sequence in context: A190432 A197563 A022138 * A028274 A272723 A245177

Adjacent sequences: A226190 A226191 A226192 * A226194 A226195 A226196

Keyword

nonn,hard,mult

Author

W. Edwin Clark, May 30 2013

Extensions

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

Status

approved