Kronecker classes of field extensions of small degree

Praeger, Cheryl E.

doi:10.1017/s1446788700032766

Cited by 11 publications

(5 citation statements)

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“…This more general situation arises for instance investigating small cliques in derangement graphs of permutation groups [67] and also in the reduction theorem in [40] for investigating arbitrary finite groups having small normal covering number, a similar reduction theorem appears also in [73]. Another typical example where this more general situation is important is the study of Kronecker classes in number fields, see [70,71,72,73,76]. Definition 1.2.…”

Section: Introductionmentioning

confidence: 99%

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Normal $2$-coverings of the finite simple groups and their generalizations

Bubboloni¹,

Spiga²,

Weigel³

2022

Preprint

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Given a finite group G, we say that G has weak normal covering number γw(G) if γw(G) is the smallest integer with G admitting proper subgroups H 1 , . . . , H γw (G) such that each element of G has a conjugate in H i , for some i ∈ {1, . . . , γw(G)}, via an element in the automorphism group of G.We prove that the weak normal covering number of every non-abelian simple group is at least 2 and we classify the non-abelian simple groups attaining 2. As an application, we classify the non-abelian simple groups having normal covering number 2. We also show that the weak normal covering number of an almost simple group is at least two up to one exception.We determine the weak normal covering number and the normal covering number of the almost simple groups having socle a sporadic simple group. Using similar methods we find the clique number of the invariably generating graph of the almost simple groups having socle a sporadic simple group.

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Section: Introductionmentioning

confidence: 99%

“…Weak normal 1-coverings of finite groups have already appeared in the literature a few times in the study of Kronecker classes [70,71,72,73]. It is still open an interesting question of Neumann and Praeger [59,Problem 11.71].…”

Section: Introductionmentioning

confidence: 99%

Normal $2$-coverings of the finite simple groups and their generalizations

Bubboloni¹,

Spiga²,

Weigel³

2022

Preprint

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“…More specifically, N KÂk K* is almost contained in N LÂk L* (see the definition below) iff P G (H K ) P G (H L ). Similar group theoretic conditions were introduced to investigate relative Brauer groups ( [3,6]), Kronecker equivalence ( [8,9,16,15,10]), equality of zeta functions ( [14]), and more recently the same group theoretic condition was shown to be equivalent to the so-called weakly Kronecker equivalence relation of algebraic number fields ( [12,11,7]). The inclusion (N KÂk .…”

Section: Introductionmentioning

confidence: 99%

Criterion for the Equality of Norm Groups of Idele Groups of Algebraic Number Fields

Stern

1997

Journal of Number Theory

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“…This notion has been studied by various authors either in different contexts or purely from the group theoretic point of view; see [4], [3], [1,Section 19.5], [2], [8], [10]. In section 2.1 we construct a finite group G with two non-conjugate subgroups U and V which are Kronecker conjugate.…”

Section: Introductionmentioning

confidence: 99%

An infinite series of Kronecker conjugate polynomials

Müller

1997

Proc. Amer. Math. Soc.

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Abstract. Let K be a field of characteristic 0, t a transcendental over K, and Γ be the absolute Galois group of K(t). Then two non-constant polynomials f, g ∈ K[X] are said to be Kronecker conjugate if an element of Γ fixes a root of f (X) − t if and only if it fixes a root of g(X) − t. If K is a number field, and f, g ∈ O K [X] where O K is the ring of integers of K, then f and g are Kronecker conjugate if and only if the value set f (O K ) equals g(O K ) modulo all but finitely many non-zero prime ideals of O K . In 1968 H. Davenport suggested the study of this latter arithmetic property. The main progress is due to M. Fried, who showed that under certain assumptions the polynomials f and g differ by a linear substitution. Further, he found non-trivial examples where Kronecker conjugacy holds. Until now there were only finitely many known such examples. This paper provides the first infinite series. The main part of the construction is group theoretic.

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Kronecker classes of field extensions of small degree

Cited by 11 publications

References 10 publications

Normal $2$-coverings of the finite simple groups and their generalizations

Normal $2$-coverings of the finite simple groups and their generalizations

Criterion for the Equality of Norm Groups of Idele Groups of Algebraic Number Fields

An infinite series of Kronecker conjugate polynomials

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