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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
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 A319788 A089498
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 March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)