login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 8 19:05 EST 2016. Contains 278948 sequences.