On groups with every normal subgroup transitive or semiregular

Bereczky, Áron; Maróti, Attila

doi:10.1016/j.jalgebra.2007.10.014

Cited by 15 publications

(151 citation statements)

References 7 publications

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“…Since |G e /G [1] xy | = 8, we have G e /G [1] xy ∼ = Dih (8). The previous paragraph shows that for any prime r = 2, we have Syl r (G x ∩ G y ) = Syl r (G [1] x ). Taking S ∈ Syl r (G [1] x ), the Frattini argument gives G x = N Gx (S)G [1] x and G e = N Ge (S) (G x ∩ G y ).…”

Section: Amalgam Methodsmentioning

confidence: 83%

“…By Tutte's Theorem, we may assume that n 2. We define G [1] x to be the kernel of the action of G x on Γ(x) and…”

Section: Amalgam Methodsmentioning

confidence: 99%

“…Frequently, we will use arc-transitivity to apply statements proved about groups like G x , G e and G xy to their conjugates (for which we use the same notation as established above). Since we wish to bound the order of G x , it is no loss to assume that G [1]…”

Section: Amalgam Methodsmentioning

confidence: 99%

“…Moreover, there exists t ∈ G e \ (G x ∩ G y ) which swaps G [1] x and G [1] y . Since |G e /G [1] xy | = 8, we have G e /G [1] xy ∼ = Dih (8). The previous paragraph shows that for any prime r = 2, we have Syl r (G x ∩ G y ) = Syl r (G [1] x ).…”

Section: Amalgam Methodsmentioning

confidence: 99%

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A class of semiprimitive groups that are graph‐restrictive

Giudici

Morgan

2014

Bulletin of London Math Soc

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Abstract. We prove that an infinite family of semiprimitive groups are graph-restrictive. This adds to the evidence for the validity of the PSV Conjecture and increases the minimal imprimitive degree for which this conjecture is open to 12. Our result can be seen as a generalisation of the well-known theorem of Tutte on cubic graphs. The proof uses the amalgam method, adapted to this new situation.

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Section: Amalgam Methodsmentioning

confidence: 83%

“…By Tutte's Theorem, we may assume that n 2. We define G [1] x to be the kernel of the action of G x on Γ(x) and…”

Section: Amalgam Methodsmentioning

confidence: 99%

Section: Amalgam Methodsmentioning

confidence: 99%

Section: Amalgam Methodsmentioning

confidence: 99%

See 2 more Smart Citations

A class of semiprimitive groups that are graph‐restrictive

Giudici

Morgan

2014

Bulletin of London Math Soc

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“…Let M be a maximal such normal subgroup of G, let Σ be the set of M -orbits, and let m = |M |. Since G is semiprimitive and M is intransitive, we have G Σ ∼ = G/M by [5,Lemma 2.4]. We now show that G Σ is quasiprimitive.…”

Section: Proof and Examples For Theorem 18mentioning

confidence: 83%

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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The distinguishing number of quasiprimitive and semiprimitive groups

2019

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The distinguishing number of G Sym(Ω) is the smallest size of a partition of Ω such that only the identity of G fixes all the parts of the partition. Extending earlier results of Cameron, Neumann, Saxl and Seress on the distinguishing number of finite primitive groups, we show that all imprimitive quasiprimitive groups have distinguishing number two, and all non-quasiprimitive semiprimitive groups have distinguishing number two, except for GL(2, 3) acting on the eight non-zero vectors of F 2 3 , which has distinguishing number three.

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On groups with every normal subgroup transitive or semiregular

Abstract: We investigate finite solvable permutation groups in which all normal subgroups are transitive or semiregular. The motivation comes from universal algebra: such groups are examples of collapsing transformation monoids.

Cited by 15 publications

References 7 publications

A class of semiprimitive groups that are graph‐restrictive

A class of semiprimitive groups that are graph‐restrictive

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

The distinguishing number of quasiprimitive and semiprimitive groups

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