A002106 Number of transitive permutation groups of degree n.

1, 1, 2, 5, 5, 16, 7, 50, 34, 45, 8, 301, 9, 63, 104, 1954, 10, 983, 8, 1117, 164, 59, 7, 25000, 211, 96, 2392, 1854, 8, 5712, 12, 2801324, 162, 115, 407, 121279, 11, 76, 306, 315842, 10, 9491, 10, 2113, 10923, 56, 6

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• core
• hard
• more
• nice

Comments

It is conjectured that this is the number of Galois groups for irreducible polynomials of order n. (All such Galois groups are transitive.) - Charles R Greathouse IV, May 28 2014

Let G be a transitive permutation groups of degree n. Then G occurs as a Galois group for an irreducible polynomial of degree n with coefficients K if and only if K admits a Galois extension with Galois group G. ("=>" is true by definition of the Galois group for an irreducible polynomial; for "<=", see user631's answer in the Math Overflow link). Hence the conjecture above is equivalent to the inverse Galois problem. Every finite group can be realized as a Galois group of some extension L/K, but for a fixed base field K (for example, K = Q is the field of rational numbers) the question is usually open. - Jianing Song, May 26 2025

Examples

			a(3)=2: A_3 and S_3.
		

References

• G. Butler and J. McKay, personal communication.
• C. C. Sims, Computational methods in the study of permutation groups, pp. 169-183 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Marina Anagnostopoulou-Merkouri, R. A. Bailey, and Peter J. Cameron, Permutation groups, partition lattices and block structures, arXiv:2409.10461 [math.GR], 2024. See Table 1 p. 45.
• G. Butler and J. McKay, The transitive groups of degree up to eleven, Comm. Algebra, 11 (1983), 863-911.
• G. Butler and J. McKay, The transitive groups of degree up to eleven, Comm. Algebra, 11 (1983), 863-911. [Annotated scanned copy]
• John J. Cannon and Derek F. Hol, The transitive permutation groups of degree 32
• F. N. Cole, Note on the substitution groups of six, seven, and eight letters, Bull. Amer. Math. Soc. 2 (1893), 184-190. Gives a(8)=48 instead of 50.
• Computational Algebra Group, Summary of New Features in Magma V2.21
• J. Conway, A. Hulpke, and J. McKay, On Transitive Permutation Groups, LMS Journal of Computation and Mathematics 1 (1998), pp. 1-8. See especially Appendix A.
• D. Holt, Enumerating subgroups of the symmetric group, in Computational Group Theory and the Theory of Groups, II, edited by L.-C. Kappe, A. Magidin and R. Morse. AMS Contemporary Mathematics book series, vol. 511, pp. 33-37. [Annotated copy]
• Derek Holt and Gordon Royle, A Census of Small Transitive Groups and Vertex-Transitive Graphs, arXiv:1811.09015 [math.CO], 2018.
• A. Hulpke, Transitive groups of small degree
• A. Hulpke, Konstruktion transitiver Permutationsgruppen, Dissertation, RWTH Aachen, 1996.
• A. Hulpke, Constructing transitive permutation groups, J. Symbolic Comput. 39 (2005), 1-30.
• E. G. Köhler, F. H. Lutz, Triangulated manifolds with few vertices: Vertex-transitive triangulations, arXiv:math/0506520 [math.GT], 2005.
• J. Labelle and Y. N. Yeh, The relation between Burnside rings and combinatorial species, J. Combin. Theory, A 50 (1989), 269-284. See page 280.
• Math Overflow, What (permutation) groups can occur as galois groups of irreducible polynomials of degree n
• G. A. Miller, On the lists of all the substitution groups that can be formed with a given number of elements, Bull. Amer. Math. Soc., 2 (1896), 138-145.
• Wikipedia, Inverse Galois problem
• Index entries for sequences related to groups
• Index entries for "core" sequences

Crossrefs

Cf. A000001, A000019, A177244, A186277.

Programs

GAP

a:=function(n)
return Length(AllTransitiveGroups(NrMovedPoints,n));
end; # Charles R Greathouse IV, May 28 2014

Extensions

Corrected and extended to degree 31 by Alexander Hulpke, Aug 15 1996

Further corrections from Alexander Hulpke, Feb 19 2002

Degree 32 extended by Artur Jasinski, Feb 17 2011

Extended to degree 47 by Gabriel Verret, May 07 2016