Characters and generation of Sylow 2-subgroups

Navarro, Gabriel; Rizo, Noelia; Fry, A. A. Schaeffer; Vallejo, Carolina

doi:10.1090/ert/555

Cited by 10 publications

(7 citation statements)

References 34 publications

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“…In the first part of the present paper we consider groups G with |P : Φ(P )| = p 2 , i. e. P is generated by two elements, but not by one. For p = 2 this property is detectable by X(G) as was shown by Navarro et al [23]. We obtain the corresponding result for odd primes p provided that G is p-constrained in Corollary 5.…”

Section: Introductionsupporting

confidence: 83%

Groups with 2-generated Sylow subgroups and their character tables

Moretó¹,

Sambale²

2022

Preprint

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Let G be a finite group with Sylow p-subgroup P . We show that the character table of G determines whether P has maximal nilpotency class and whether P is a minimal non-abelian group. The latter result is obtained from a precise classification of the corresponding groups G in terms of their composition factors. For p-constrained groups G we prove further that the character table determines whether P can be generated by two elements.

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Section: Introductionsupporting

confidence: 83%

Groups with 2-generated Sylow subgroups and their character tables

Moretó¹,

Sambale²

2022

Preprint

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“…The proof of one direction of the Brauer Height Zero Conjecture in [KM13] and the recent proof for principal blocks [MN21] have been major breakthroughs in this line of investigation. Other recent results studying the p-structural properties of the defect group in terms of properties of the irreducible characters in the block can be found in [FLLMZ19], [GRSS20] and [NRSV21]. The present paper is a contribution to this lively area of research.…”

Section: Introductionmentioning

confidence: 68%

Character degrees in blocks and defect groups

Giannelli¹,

Martínez²,

Fry³

2021

Preprint

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A recent question of Gabriel Navarro asks whether it is true that the derived length of a defect group is less than or equal to the number of degrees of irreducible characters in a block. In this article, we bring new evidence towards the validity of this statement. Reduction to simple groups2.1. Auxiliary lemmas.Lemma 2.1. Let G be a finite group, N Ÿ G, and θ P IrrpB 0 pNqq. Assume θ extends to χ P IrrpB 0 pGqq. Then tµχ | µ P IrrpB 0 pG{Nqqu Ď IrrpG|θq X IrrpB 0 pGqq.Proof. By Gallagher's correspondence, IrrpG|θq " tµχ | µ P IrrpG{Nqu. We have χ " 1 G{N χ and then by [Riz18, Lemma 2.4], if µ P IrrpB 0 pG{Nqq then µχ P IrrpB 0 pGqq. Lemma 2.2. Let G be a finite group and let N " S 1 ˆ¨¨¨ˆS t Ÿ G, where the S i 's are subgroups permuted by G. Let S " S 1 . Assume there exists α P IrrpN G pSq{C G pSqq in the principal block and such that S is not contained in kerpαq. Then α G P IrrpB 0 pGqq.

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“…Notice that a character χ ∈ Irr(G) is 2-rational if, and only if, χ is G-fixed. The set of 2-height zero characters fixed under the action of σ 1 has been recently studied in connection with the number of generators of 2-defect groups (see [RSV,NRSV,Val1]).…”

Section: Theorem a For Quasisimple Groupsmentioning

confidence: 99%

The fields of values of characters of degree not divisible byp

Navarro

Tiep

2021

Forum of Mathematics, Pi

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We study the fields of values of the irreducible characters of a finite group of degree not divisible by a prime p. In the case where $p=2$ , we fully characterise these fields. In order to accomplish this, we generalise the main result of [ILNT] to higher irrationalities. We do the same for odd primes, except that in this case the analogous results hold modulo a simple-to-state conjecture on the character values of quasi-simple groups.

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Characters and generation of Sylow 2-subgroups

Cited by 10 publications

References 34 publications

Groups with 2-generated Sylow subgroups and their character tables

Groups with 2-generated Sylow subgroups and their character tables

Character degrees in blocks and defect groups

The fields of values of characters of degree not divisible byp

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