cp's OEIS Frontend

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

%I A260645 #24 Nov 10 2021 07:31:32
%S A260645 1,1,5,19,19,5,41,385,231,19,109,95,155,41,95,41387,271,231,341,361,
%T A260645 205,109,505,1925,3337,155,36118,779,811,95,929,19823665,545,271,779,
%U A260645 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
%N A260645 The number of central quasigroups (also known as T-quasigroups, or quasigroups affine over an abelian group) of order n, up to isomorphism.
%C A260645 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.
%H A260645 David Stanovsky, <a href="/A260645/b260645.txt">Table of n, a(n) for n = 1..63</a>
%H A260645 David Stanovský and Petr Vojtechovský, <a href="http://arxiv.org/abs/1511.03534">Central and medial quasigroups of small order</a>, arxiv preprint arXiv:1511.03534 [math.GR], 2015.
%o A260645 (GAP) # gives the number of central quasigroups over SmallGroup(n, k)
%o A260645 LoadPackage("loops");
%o A260645 CQ := function( n, k )
%o A260645     local G, ct, elms, inv, A, f_reps, count,f, Cf, O, g_reps, g, Cfg, W, unused, c, Wc;
%o A260645     G := SmallGroup( n, k );
%o A260645     G := IntoLoop( G );
%o A260645     ct := CayleyTable( G );
%o A260645     elms := Elements( G );
%o A260645     inv := List( List( [1..n], i -> elms[i]^(-1) ), x -> x![1] );
%o A260645     A := AutomorphismGroup( G );
%o A260645     f_reps := List( ConjugacyClasses( A ), Representative );
%o A260645     count := 0;
%o A260645     for f in f_reps do
%o A260645         Cf := Centralizer( A, f );
%o A260645         O := OrbitsDomain( Cf, A );
%o A260645         g_reps := List( O, x -> x[1] );
%o A260645         for g in g_reps do
%o A260645             Cfg := Intersection( Cf, Centralizer( A, g ) );
%o A260645             W := Set( [1..n], w -> ct[w][ inv[ ct[w^f][w^g] ] ] );
%o A260645             unused := [1..n];
%o A260645             while not IsEmpty( unused ) do
%o A260645                 c := unused[1];
%o A260645                 count := count + 1;
%o A260645                 if Size(W) = Length(unused) then
%o A260645                     unused := [];
%o A260645                 else
%o A260645                     Wc := Set( W, w -> ct[w][c] );
%o A260645                     Wc := Union( Orbits( Cfg, Wc ) );
%o A260645                     unused := Difference( unused, Wc );
%o A260645                 fi;
%o A260645             od;
%o A260645         od;
%o A260645     od;
%o A260645     return count;
%o A260645 end;
%Y A260645 Cf. A226193.
%K A260645 nonn,hard,mult
%O A260645 1,3
%A A260645 _David Stanovsky_, Nov 12 2015