cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-17 of 17 results.

A124937 Number of solvable transitive Galois groups for polynomials of degree n.

Original entry on oeis.org

1, 1, 2, 5, 3, 12, 4, 45, 30, 24, 4, 265, 6, 36, 64, 1905, 5, 892, 6, 759, 108, 32, 4, 24193, 132, 70, 2328, 1237, 6, 3816, 8
Offset: 1

Views

Author

Artur Jasinski, Nov 13 2006

Keywords

Examples

			a(5) = 3: for polynomials of degree 5 we have 3 solvable groups: C5 (T5_1), D5 (T5_2) and F5(T5_3)

Links

Crossrefs

Cf. A002106, A124938.

Programs

  • GAP
    "a(15)= "; l:=AllTransitiveGroups(NrMovedPoints,15,IsSolvable,true); # Artur Jasinski, Feb 04 2007
  • Magma
    // a(10)
for g in [1..45] do
G:=TransitiveGroup(10,g);
IsSolvable(G);
end for;

Extensions

More terms from Artur Jasinski, Feb 04 2007

A198342 Number of non-solvable transitive permutation groups for polynomials of degree n.

Original entry on oeis.org

0, 0, 0, 0, 2, 4, 3, 5, 4, 21, 4, 36, 3, 27, 40, 49, 5, 91, 2, 358, 56, 27, 3, 807, 79, 26, 64, 617, 2, 1896, 4
Offset: 1

Views

Author

Artur Jasinski, Oct 23 2011

Keywords

Comments

For prime degrees of polynomials see A201443.
All non-solvable groups are non-commutative.
Is this the same as A124938 ? - R. J. Mathar, Oct 04 2018

Examples

			a(4)=0 because for quartic polynomials all groups are solvable.
a(5)=2 because for quintic polynomials we have two non-solvable groups: A(5) and S(5).

Links

Crossrefs

Cf. A002106, A131932.

Programs

  • Magma
    // for a(16):
for g in [1..1954] do
G:=TransitiveGroup(16,g);
IsSolvable(G);
end for

A201443 Number of non-solvable transitive permutation groups for polynomials of degree p(n), where p(n) is n-th prime.

Original entry on oeis.org

0, 0, 2, 3, 4, 3, 5, 2, 3, 2, 4
Offset: 1

Views

Author

Artur Jasinski, Dec 01 2011

Keywords

Links

Crossrefs

Cf. A002106, A131932, A198342.

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

Original entry on oeis.org

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

Views

Author

John Erickson and Alexander Hulpke, Nov 21 2020

Keywords

Comments

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.

Examples

			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) ] )

Links

Crossrefs

Cf. A005432, A002106.

Programs

  • GAP
    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

A132221 Number of imprimitive transitive permutation groups of degree n.

Original entry on oeis.org

0, 0, 0, 3, 0, 12, 0, 43, 23, 36, 0, 295, 0, 59, 98, 1932, 0, 979, 0, 1113, 155, 55, 0, 24995, 183, 89, 2377, 1840, 0, 5708, 0, 2801317, 158, 113, 401, 121257, 0, 72, 304, 315834, 0, 9487, 0, 2109, 10914, 54, 0
Offset: 1

Views

Author

Artur Jasinski, Aug 14 2007

Keywords

Comments

Smallest degree of an imprimitive group is 4. The groups of degree 4 are C_4, V_4, D_4.
Imprimitive groups do not exist with prime degrees.

Crossrefs

Cf. A000019, A002106.

Formula

a(n) = A002106(n) - A000019(n).

Extensions

More terms from Vaclav Kotesovec, Jul 18 2022

A185176 a(n) = maximal number of different Galois groups with that same order for polynomials of degree n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 8, 4, 5, 1, 30, 1, 5, 5, 260, 1, 43, 1, 57, 7, 4, 1, 1930, 8, 10, 99, 93, 1, 223, 1
Offset: 2

Views

Author

Artur Jasinski, Feb 19 2011

Keywords

Comments

For prime p, a(p)=1.
For nonprime n, the most frequently seen orders are:
4 = 4,
6 = 24,
8 = 32,
9 = 54,
10 = 200,
12 = 192,
14 = 2688,
15 = 360,
16 = 256,
18 = 1296,
20 = {5120,40000},
21 = 30618,
22 = 2420,
24 = 1536,
25 = {500,2500,12500},
26 = 4056,
27 = 4374,
28 = 114688,
30 = 24000000

Examples

			a(4)=2 because for polynomials of degree 4, there are two different groups of order 4.
a(20)=57 because for polynomials of degree 20, there are 57 different groups of order 5120 and 57 different groups of order 40000.

Crossrefs

Cf. A002106, A177244, A186277, A186306, A186307, A186308.

A322100 Number of minimal transitive permutation groups of degree n.

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 5, 2, 6, 1, 17, 1, 6, 4, 75, 1, 23, 1, 47, 5, 6, 1, 213, 2, 7, 20, 30, 1, 79, 1, 12033, 3, 7, 4, 436, 1, 5, 4, 1963, 1, 84, 1, 148, 41, 4, 1
Offset: 1

Views

Author

Danny Rorabaugh, Nov 26 2018

Keywords

Comments

A transitive group is minimal provided it has no proper transitive subgroups.

Examples

			There are two transitive groups of degree 3, A_3 and S_3, so A002106(3)=2. However, a(3)=1, because A_3 is minimal, but S_3 has proper transitive subgroups A_3 and S_2.

Links

Crossrefs

Cf. A002106.
Previous Showing 11-17 of 17 results.

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