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A208235 Maximal order of a subgroup of the symmetric group of degree n that contains no 3-cycle. 1
1, 2, 2, 8, 20, 120, 168, 1344, 1512, 3840, 7920, 95040, 95040 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

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

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

EXAMPLE

A Sylow 2-subgroup of S_4 is of order 8, and contains no 3-cycle. The only subgroups of S_4 with more than 8 elements are A_4 and S_4, which both contain 3-cycles. So a(4) = 8.

PROG

(GAP)

Has3:=function(G, n)

local x, p;

for p in Elements(G) do

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

  if  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 Has3(h, n) = false then

    nn:=Size(h);

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

  fi;

od;;

return MM;

end;;

CROSSREFS

Cf. A208232.

Sequence in context: A178076 A137774 A167532 * A151377 A151407 A130102

Adjacent sequences:  A208232 A208233 A208234 * A208236 A208237 A208238

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 18 06:39 EDT 2014. Contains 240706 sequences.