On Transitive Permutation Groups

Conway, John H.; Hulpke, Alexander; McKay, John

doi:10.1112/s1461157000000115

Cited by 65 publications

(71 citation statements)

References 6 publications

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“…Expanding this out, we get a polynomial of degree eight, with 0 as the trace term. The coefficient of degree five has the form of the resolvent for L (3,2), which consists of the seven products of three roots, corresponding to the lines of the projective plane of order 2. If we label the nonzero elements of the 2-elementary group by integers from 1 to 7, then in one of the labelings we obtain…”

Section: Semidirect Productsmentioning

confidence: 99%

“…We will use the notation T i to denote the ith transitive group of degree n (with n understood from context) in the tables of [1], but also what will probably become the new standard: a naming scheme for permutation groups given in [2]. This should be consulted for information about the meaning of these names.…”

Section: Introductionmentioning

confidence: 99%

“…If we now substitute for b j the roots of a polynomial contained in L (3,2), with the roots ordered according to the order determined by considering the degree six term of this new polynomial, we get a polynomial of degree eight which has roots that are sums of square roots of products of the roots b j of our degree seven polynomial, and which gives the split extension of the degree seven group by the 2-elementary group of degree eight. (This polynomial is long and complicated, and so we do not give it here; however, it is not too difficult to compute using a computer algebra package.…”

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confidence: 99%

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Some polynomials over $\mathbb{Q}(t)$ and their Galois groups

Smith

1999

Math. Comp.

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Section: Semidirect Productsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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Some polynomials over $\mathbb{Q}(t)$ and their Galois groups

Smith

1999

Math. Comp.

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“…In Ngo [6] non-regular metabelian minimally transitive groups are investigated, and Miller-Praeger [5] mention such groups in the context of vertex transitive graphs which are not Cayley graphs. A list of minimally transitive groups up to degree 30 is available in Hulpke [2], see also Conway, Hulpke and McKay [1].…”

Section: Introductionmentioning

confidence: 99%

On solvable minimally transitive permutation groups

Volta

Siemons

2007

Des. Codes Cryptogr.

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“…However, to avoid ambiguity, particularly for group extensions and semidirect products, we also make use of two standard group numberings. For Galois groups, we also use the notation nT d to denote the dth transitive subgroup of S n in the numbering of [BM83,CHM98]. For small groups, at times we use the numbering system employed by the small group library of gap [GAP06].…”

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confidence: 99%

Number fields with solvable Galois groups and small Galois root discriminants

Jones

Wallington

2012

Math. Comp.

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Abstract. We apply class field theory to compute complete tables of number fields with Galois root discriminant less than 8πe γ . This includes all solvable Galois groups which appear in degree less than 10, groups of order less than 24, and all dihedral groups D p where p is prime.Many people have studied questions of constructing complete lists of number fields subject to conditions on degree and possibly Galois group, with a goal of determining complete lists of such fields with discriminant less than a fixed bound. This study can be phrased in those terms, with the principle distinction being that our discriminant bounds apply to Galois fields, rather than to particular stem fields.For a finite group G and bound B > 0, let K(G, B) be the set of number fields in C which are Galois over Q and which have root discriminant ≤ B. It is a classical theorem that each set K(G, B) is finite. Jones and Roberts [JR07a] considered these sets with B = Ω = 8πe γ ≈ 44.7632, a constant introduced by Serre [Ser86]. This number is the asymptotic limit for root discriminant bounds assuming the Generalized Riemann Hypothesis. Jones and Roberts conjecture that there are only finitely many such fields in the union of the K(G, Ω), with G running through all finite groups. They determine all sets K(A, Ω) where A is an abelian group, and most sets K(G, Ω) for groups G which appear as Galois groups of irreducible polynomials of degree at most 6. Here we extend those results using class field theory. We complete their study of Galois groups which appear in degree 6, treat all solvable extensions in degrees 7-9, as well as for all groups G with |G| ≤ 23. Finally, we determine the sets K(D , Ω) where is an odd prime.The primary theoretical difficulty in carrying out these computations is to deduce bounds which effectively screen out conductors for our extensions. Section 1 sets notation and describes some of the methods used before Section 2 addresses this question, and describes our overall process. Section 3 summarizes the results of our computations.

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On Transitive Permutation Groups

Abstract: We assign names and new generators to the transitive groups of degree up to 15, reflecting their structure.

Cited by 65 publications

References 6 publications

Some polynomials over $\mathbb{Q}(t)$ and their Galois groups

Some polynomials over $\mathbb{Q}(t)$ and their Galois groups

On solvable minimally transitive permutation groups

Number fields with solvable Galois groups and small Galois root discriminants

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