The Primitive Permutation Groups of Degree Less Than 4096

Coutts, Hannah Jane; Quick, Martyn; Roney-Dougal, Colva M.

doi:10.1080/00927872.2010.515521

Cited by 28 publications

(33 citation statements)

References 27 publications

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“…In permutation group of degree r dividing n/m. By Theorem 1.3, c(G Σ ) 8 3 log 2 r − 4 3 . As |M | = m 2, the desired bound is proved as follows:…”

Section: Proofs Of Theorems 13 and 16mentioning

confidence: 91%

“…Finally, we assume that G is primitive. We used a database of primitive groups of small degree (see [8]) to check that the bound is satisfied when n 24 and equality holds only for T 1 = S 4 . We thus assume that n 25.…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

“…Since s divides p f − 1, the possibilities reduce to (s a , p f ) ∈ {(2, 3), (3, 2 2 ), (2 2 , 3), (2 3 , 3)}. Finally, using [6] we can determine the C 6 -subgroups of GL(2, 3), GL (3,4), GL(4, 3) and GL (8,3); in all these cases, the bound (1) holds strictly. This concludes the C 6 case.…”

Section: Proof Of Theorem 14mentioning

confidence: 99%

“…Henceforth assume that n 5, that G is a primitive permutation group of degree n. If G is nonaffine then, by Theorem 1.6, c(G) c na log 2 n − 4 3 . Since c na < 8 3 , Theorem 1.3 holds with a strict inequality in this case. Thus we may assume that G is of affine type, so n = p d for some prime p and integer d 1, and G = (C p ) d H where H is an irreducible subgroup of GL(d, p).…”

mentioning

confidence: 91%

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Bounding the composition length of primitive permutation groups and completely reducible linear groups

Glasby

Praeger

Rosa

et al. 2018

J. London Math. Soc.

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We obtain upper bounds on the composition length of a finite permutation group in terms of the degree and the number of orbits, and analogous bounds for primitive, quasiprimitive and semiprimitive groups. Similarly, we obtain upper bounds on the composition length of a finite completely reducible linear group in terms of some of its parameters. In almost all cases we show that the bounds are sharp, and describe the extremal examples.Problem 1.7. Which other infinite families of permutation groups have composition lengths bounded above by a logarithmic function of the degree?Our final main result gives examples of two such families. A permutation group is quasiprimitive if each of its nontrivial normal subgroups is transitive. It is semiprimitive if each of its normal subgroups is either semiregular or transitive. (A permutation group is semiregular if the only element fixing a point is the identity.) Theorem 1.8. Let G be a permutation group of degree n.(a) If G is quasiprimitive but not primitive, thenwhere c na = 10 3 log 2 5 = 1.43 · · · as in Theorem 1.6. (b) If G is semiprimitive but not quasiprimitive, then c(G) 8 3 log 2 n − 3.We give infinitely many examples to show that the bound in Theorem 1.8(b) is best possible (see Example 6.2). For a semiprimitive group G, a normal subgroup of G which is minimal † The statement in [24, Theorem 2.10] refers to a paper 'in preparation' (reference [Py5] in [24]).

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“…In permutation group of degree r dividing n/m. By Theorem 1.3, c(G Σ ) 8 3 log 2 r − 4 3 . As |M | = m 2, the desired bound is proved as follows:…”

Section: Proofs Of Theorems 13 and 16mentioning

confidence: 91%

Section: Proof Of Theorem 12mentioning

confidence: 99%

Section: Proof Of Theorem 14mentioning

confidence: 99%

mentioning

confidence: 91%

See 2 more Smart Citations

Bounding the composition length of primitive permutation groups and completely reducible linear groups

Glasby

Praeger

Rosa

et al. 2018

J. London Math. Soc.

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show abstract

“…The complexity of this algorithm grows with n and q. More recently, H. J. Coutts, M. Quick, and C. Roney-Dougal [5] classified the insoluble irreducible subgroups of GL(n, p) for p n < 4096.…”

Section: Related Workmentioning

confidence: 99%

L. g. kovács and Linear Groups

Detinko¹,

Flannery²

2016

J. Aust. Math. Soc.

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On symmetric designs with flag‐transitive and point‐quasiprimitive automorphism groups

Zhang,

Chen,

Zhou

2023

J of Combinatorial Designs

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Let be a nontrivial symmetric ‐design with , and let be a flag‐transitive automorphism group of . In this paper, we show that if is quasiprimitive on , then is of holomorph affine or almost simple type. Moreover, if is imprimitive on , then is of almost simple type. According to this observation and to the classification of the finite simple groups we determine all such symmetric designs and the corresponding automorphism groups. We conclude with two open problems and a conjecture.

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The Primitive Permutation Groups of Degree Less Than 4096

Abstract: In this paper we use the Classification of the Finite Simple Groups, the O'Nan-Scott Theorem and Aschbacher's theorem to classify the primitive permutation groups of degree less than 4096. The results will be added to the primitive groups databases of GAP and Magma.

Cited by 28 publications

References 27 publications

Bounding the composition length of primitive permutation groups and completely reducible linear groups

Bounding the composition length of primitive permutation groups and completely reducible linear groups

L. g. kovács and Linear Groups

On symmetric designs with flag‐transitive and point‐quasiprimitive automorphism groups

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