cp's OEIS Frontend

A174511 The number of isomorphism classes of subgroups of the symmetric group S_n.

%I A174511 #26 Jan 14 2024 06:56:31
%S A174511 1,2,4,9,16,29,55,137,241,453,894,2065,3845
%N A174511 The number of isomorphism classes of subgroups of the symmetric group S_n.
%C A174511 Two subgroups are considered to be isomorphic here if they are isomorphic as abstract groups, not as permutation groups. - _N. J. A. Sloane_, Nov 28 2010
%H A174511 A. Distler and T. Kelsey, <a href="http://arxiv.org/abs/1301.6023">The semigroups of order 9 and their automorphism groups</a>, arXiv preprint arXiv:1301.6023 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 19 2013
%H A174511 J. Schmidt, <a href="http://math.stackexchange.com/questions/76176/enumerating-all-subgroups-of-the-symmetric-group">Enumerating all subgroups of the symmetric group.</a>
%e A174511 a(3) = 4 since S_3 contains up to isomorphism exactly one subgroup of each of the orders 1,2,3,6.
%o A174511 (GAP)
%o A174511 a:=[];
%o A174511 for n in [1,2,3,4,5,6,7,8,9,10] do
%o A174511   G:=SymmetricGroup(n);
%o A174511   R:=ConjugacyClassesSubgroups(G);
%o A174511   RR:=ListX(R,Representative);
%o A174511   T:=[RR[1]];
%o A174511   for g in RR do
%o A174511     flag:=false;
%o A174511     for h in T do
%o A174511       if IsomorphismGroups(g,h)<>fail then
%o A174511         flag:=true;
%o A174511         break;
%o A174511       fi;
%o A174511     od;
%o A174511     if flag=false then Add(T,g); fi;
%o A174511   od;
%o A174511   Add(a,Size(T));
%o A174511 od;
%o A174511 Print(a,"\n");
%Y A174511 Cf. A000638, A005432.
%K A174511 nonn,more
%O A174511 1,2
%A A174511 _W. Edwin Clark_, Nov 28 2010
%E A174511 a(11) and a(12) from _Stephen A. Silver_, Feb 24 2013
%E A174511 a(13) (as calculated by Jack Schmidt) from _L. Edson Jeffery_, May 26 2013