Cyclic Quotients of Transitive Groups

Guralnick, Robert M.

doi:10.1006/jabr.2000.8540

Cited by 5 publications

(3 citation statements)

References 18 publications

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“…However, if we restrict to faithful subdegrees of a transitive group G, that is, subdegrees d such that there exists an orbit of length d of a stabilizer G α on which G α acts faithfully, then in fact we can show that a conclusion analogous to the statement of Theorem 1.7 does hold. We note that, in particular, every primitive permutation group has a faithful subdegree [12,Theorem 3]. Theorem 1.14.…”

Section: 2mentioning

confidence: 99%

Coprime subdegrees for primitive permutation groups and completely reducible linear groups

et al. 2012

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Abstract. In this paper we answer a question of Gabriel Navarro about orbit sizes of a finite linear group H ⊆ GL(V ) acting completely reducibly on a vector space V : if the H-orbits containing the vectors a and b have coprime lengths m and n, we prove that the H-orbit containing a + b has length mn. Such groups H are always reducible if n, m > 1. In fact, if H is an irreducible linear group, we show that, for every pair of non-zero vectors, their orbit lengths have a non-trivial common factor.In the more general context of finite primitive permutation groups G, we show that coprime non-identity subdegrees are possible if and only if G is of O'Nan-Scott type AS, PA or TW. In a forthcoming paper we will show that, for a finite primitive permutation group, a set of pairwise coprime subdegrees has size at most 2. Finally, as an application of our results, we prove that a field has at most 2 finite extensions of pairwise coprime indices with the same normal closure.

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Section: 2mentioning

confidence: 99%

Coprime subdegrees for primitive permutation groups and completely reducible linear groups

et al. 2012

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“…Guralnick [7] recently obtained a more precise version of Cantor's result for cyclotomic extensions (see also [1] and [11] for other results in abelian extensions). For fields, it states that if L/k is cyclic and L is contained in the Galois closure of k(α) over k, then…”

Section: Resultsmentioning

confidence: 94%

On Subfields of a Field Generated by Two Conjugate Algebraic Numbers

Drungilas¹,

Dubickas²

2004

Proceedings of the Edinburgh Mathematical Society

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Let k be a field, and let α and α be two algebraic numbers conjugate over k. We prove a result which implies that if L ⊂ k(α, α ) is an abelian or HamiltonianThis is related to a certain question concerning the degree of an algebraic number and the degree of a quotient of its two conjugates provided that the quotient is a root of unity, which was raised (and answered) earlier by Cantor. Moreover, we introduce a new notion of the non-torsion power of an algebraic number and prove that a monic polynomial in X-irreducible over a real field and having m roots of equal modulus, at least one of which is real-is a polynomial in X m .Keywords: field; subfield; algebraic numbers; abelian and Hamiltonian extensions; roots of unity 2000 Mathematics subject classification: Primary 11R04; 11R20; 11R32; 12F10 ResultsLet k be a field, and let k a be an algebraic closure of k. In this paper we give a short and self-contained proof of the following theorem.The condition of the theorem on L ∩ k(α) always holds if L/k is an abelian extension. However, this is not the only case. We say that L is a Hamiltonian extension of k if it is a Galois extension and Gal(L/k) is a Hamiltonian group, namely, a non-abelian group every subgroup of which is normal. For example, the quaternion group Q 8 is Hamiltonian. It occurs as a Galois group over the field of rational numbers Q, since it is a 2-group. See p. 190 of [8] for the description of all Hamiltonian groups.

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“…By a theorem of Guralnick [33,Theorem 3], H acts faithfully on at least one of its orbits in Ω \ {α}. Moreover, if we assume G is extremely primitive then Manning's theorem [66,Corollary I,p.821] implies that H acts faithfully and primitively on all of its orbits in Ω \ {α}.…”

Section: Extremely Primitive Groups Let Gmentioning

confidence: 99%

The Classification of Extremely Primitive Groups

Burness

Thomas

2021

International Mathematics Research Notices

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Dedicated to the memory of Jan Saxl Let $G$ be a finite primitive permutation group on a set $\Omega $ with nontrivial point stabilizer $G_{\alpha }$. We say that $G$ is extremely primitive if $G_{\alpha }$ acts primitively on each of its orbits in $\Omega \setminus \{\alpha \}$. These groups arise naturally in several different contexts, and their study can be traced back to work of Manning in the 1920s. In this paper, we determine the almost simple extremely primitive groups with socle an exceptional group of Lie type. By combining this result with earlier work of Burness, Praeger, and Seress, this completes the classification of the almost simple extremely primitive groups. Moreover, in view of results by Mann, Praeger, and Seress, our main theorem gives a complete classification of all finite extremely primitive groups, up to finitely many affine exceptions (and it is conjectured that there are no exceptions). Along the way, we also establish several new results on base sizes for primitive actions of exceptional groups, which may be of independent interest.

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Cyclic Quotients of Transitive Groups

Cited by 5 publications

References 18 publications

Coprime subdegrees for primitive permutation groups and completely reducible linear groups

Coprime subdegrees for primitive permutation groups and completely reducible linear groups

On Subfields of a Field Generated by Two Conjugate Algebraic Numbers

The Classification of Extremely Primitive Groups

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