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User: Alexander Hulpke

Alexander Hulpke has authored 1 sequences.

A337015 Number of distinct transitive subgroups of S_n, counting conjugates as distinct.

1, 1, 2, 9, 20, 279, 512, 19087, 71602, 636365, 1517042, 321965982, 240609602, 8809543877, 144729615032, 26818608209252, 6603755558402, 2737593592637477

Offset: 1

John Erickson and Alexander Hulpke, Nov 21 2020

This sequence is the labeled version of A002106. I have proven that A005432(p)-a(p) == 1 (mod p) if p is prime. Based on n<= 18,
I have conjectured that log(A005432(n)/a(n)) > (n-1)/2 for n prime and log(A005432(n)/a(n)) < (n-1)/2 for n composite.
L. Pyber shows c^{n^2*(1+o(1))} <= a(n) <= d^{n^2*(1+o(1))}, c=2^{1/16}, d=24^{1/6}; conjectures lower bound is accurate.

			For n = 4 the following 9 subgroups of S_4 are transitive:
Group( [ (1,4)(2,3), (1,3)(2,4) ] )
Group( [ (1,3,2,4), (1,2)(3,4) ] )
Group( [ (1,4,3,2), (1,3)(2,4) ] )
Group( [ (1,2,4,3), (1,4)(2,3) ] )
Group( [ (1,4)(2,3), (1,3)(2,4), (3,4) ] )
Group( [ (1,2)(3,4), (1,3)(2,4), (1,4) ] )
Group( [ (1,2)(3,4), (1,4)(2,3), (2,4) ] )
Group( [ (1,4)(2,3), (1,3)(2,4), (2,4,3) ] )
Group( [ (1,4)(2,3), (1,3)(2,4), (2,4,3), (3,4) ] )

John Erickson, COUNTING TRANSITIVE SUBGROUPS OF Sn
L. Pyber, Enumerating Finite Groups of Given Order, Ann. Math. 137 (1993), 203-220.

Cf. A005432, A002106.

NrTransSubSn:=function(n)
local s,cnt,i,u,no;
  s:=SymmetricGroup(n);
  cnt:=0;
  for i in [1..NrTransitiveGroups(n)] do
    u:=TransitiveGroup(n,i);
    no:=Normalizer(s,u);
    cnt:=cnt+IndexNC(s,no);
    Print("Class ",i,", found ",IndexNC(s,no)," new, total: ",cnt,"\n");
  od;
  return cnt;
end; # Alexander Hulpke

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User: Alexander Hulpke

A337015 Number of distinct transitive subgroups of S_n, counting conjugates as distinct.

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