A208232 - OEIS

%I #27 Feb 15 2013 08:49:46

%S 1,1,1,4,20,120,168,1344,1512,1920,7920,95040,95040

%N Maximum order of a subgroup of the symmetric group of degree n that contains no 2-cycle and no 3-cycle.

%H MathOverflow, <a href="http://mathoverflow.net/questions/118494">Largest permutation group without 2-cycles or 3-cycles</a>

%e a(4) = 4 since the subgroups of S_4 up to conjugation as computed by GAP are:

%e H(1) =  { ()}

%e H(2) =  { (), (1,3)(2,4)}

%e H(3) =  { (), (3,4)}

%e H(4) =  { (), (2,3,4), (2,4,3)}

%e H(5) =  { (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)}

%e H(6) =  { (), (3,4), (1,2), (1,2)(3,4)}

%e H(7) =  { (), (1,2)(3,4), (1,3,2,4), (1,4,2,3)}

%e H(8) =  { (), (3,4),(2,3), (2,3,4), (2,4,3), (2,4)}

%e H(9) =  { (), (3,4), (1,2), (1,2)(3,4), (1,3)(2,4), (1,3,2,4), (1,4,2,3), (1,4)(2,3)}

%e H(10) = { (), (2,3,4), (2,4,3), (1,2)(3,4), (1,2,3), (1,2,4), (1,3,2), (1,3,4), (1,3)(2,4), (1,4,2), (1,4,3), (1,4)(2,3)}

%e H(11) = { (), (3,4), (2,3), (2,3,4), (2,4,3), (2,4), (1,2), (1,2)(3,4), (1,2,3), (1,2,3,4), (1,2,4,3), (1,2,4), (1,3,2), (1,3,4,2), (1,3), (1,3,4), (1,3)(2,4), (1,3,2,4), (1,4,3,2), (1,4,2), (1,4,3), (1,4), (1,4,2,3), (1,4)(2,3)}

%e Only H(1), H(2), H(5) and H(7) contain neither 2-cycle nor 3-cycle and the largest of these groups has order 4.

%e I use here the GAP convention of writing cycles with commas.

%o (GAP)

%o Has23:=function(G,n)

%o local x,p;

%o for p in Elements(G) do

%o   x:=Product(CycleLengths(p,[1..n]));

%o   if x = 2 or x = 3 then return true; fi;

%o od;

%o return false;

%o end;;

%o a:=function(n)

%o local MM,h,nn;

%o MM:=0;;

%o for H in ConjugacyClassesSubgroups(SymmetricGroup(n)) do

%o   h:=Representative(H);

%o   if Size(h)<=MM then continue; fi;

%o   if Has23(h,n) = false then

%o     nn:=Size(h);

%o     if nn > MM then MM:=nn; Mg:=h; fi;

%o   fi;

%o od;;

%o return MM;

%o end;;

%Y Cf. A208235.

%K nonn,more

%O 1,4

%A _W. Edwin Clark_, Jan 10 2013

%E a(10)-a(13) from _Stephen A. Silver_, Feb 14 2013