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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 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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