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

1, 2, 4, 9, 16, 29, 55, 137, 241, 453, 894, 2065, 3845

Offset: 1

Author

W. Edwin Clark, Nov 28 2010

Keywords

• nonn
• more

Comments

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

Examples

			a(3) = 4 since S_3 contains up to isomorphism exactly one subgroup of each of the orders 1,2,3,6.
		

Links

• A. Distler and T. Kelsey, The semigroups of order 9 and their automorphism groups, arXiv preprint arXiv:1301.6023 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 19 2013
• J. Schmidt, Enumerating all subgroups of the symmetric group.

Crossrefs

Cf. A000638, A005432.

Programs

GAP

a:=[];
for n in [1,2,3,4,5,6,7,8,9,10] do
  G:=SymmetricGroup(n);
  R:=ConjugacyClassesSubgroups(G);
  RR:=ListX(R,Representative);
  T:=[RR[1]];
  for g in RR do
    flag:=false;
    for h in T do
      if IsomorphismGroups(g,h)<>fail then
        flag:=true;
        break;
      fi;
    od;
    if flag=false then Add(T,g); fi;
  od;
  Add(a,Size(T));
od;
Print(a,"\n");

Extensions

a(11) and a(12) from Stephen A. Silver, Feb 24 2013

a(13) (as calculated by Jack Schmidt) from L. Edson Jeffery, May 26 2013