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A208232 Maximum order of a subgroup of the symmetric group of degree n that contains no 2-cycle and no 3-cycle. 1
1, 1, 1, 4, 20, 120, 168, 1344, 1512, 1920, 7920, 95040, 95040 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

LINKS

Table of n, a(n) for n=1..13.

MathOverflow, Largest permutation group without 2-cycles or 3-cycles

EXAMPLE

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

H(1) =  { ()}

H(2) =  { (), (1,3)(2,4)}

H(3) =  { (), (3,4)}

H(4) =  { (), (2,3,4), (2,4,3)}

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

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

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

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

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)}

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)}

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)}

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.

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

PROG

(GAP)

Has23:=function(G, n)

local x, p;

for p in Elements(G) do

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

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

od;

return false;

end;;

a:=function(n)

local MM, h, nn;

MM:=0;;

for H in ConjugacyClassesSubgroups(SymmetricGroup(n)) do

  h:=Representative(H);

  if Size(h)<=MM then continue; fi;

  if Has23(h, n) = false then

    nn:=Size(h);

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

  fi;

od;;

return MM;

end;;

CROSSREFS

Cf. A208235.

Sequence in context: A128236 A091046 A101055 * A013197 A089498 A046729

Adjacent sequences:  A208229 A208230 A208231 * A208233 A208234 A208235

KEYWORD

nonn,more

AUTHOR

W. Edwin Clark, Jan 10 2013

EXTENSIONS

a(10)-a(13) from Stephen A. Silver, Feb 14 2013

STATUS

approved

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Last modified April 24 09:53 EDT 2014. Contains 240965 sequences.