Height zero characters in principal blocks

Hưng, Nguyễn Ngọc; Fry, A. A. Schaeffer; Vallejo, Carolina

doi:10.1016/j.jalgebra.2023.01.021

Cited by 6 publications

(5 citation statements)

References 57 publications

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“…and this indeed happens for all exceptional types and all d, except the single case of type E 7 and d = 4 (see also [HSF,p. 19]).…”

Section: Principal Blocks Of Almost Simple Groups Of Lie Typementioning

confidence: 66%

Bounding p-Brauer characters in finite groups with two conjugacy classes of p-elements

Hung,

Sambale,

Tiep

2024

Isr. J. Math.

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Let k(B0) and l(B0) respectively denote the number of ordinary and p-Brauer irreducible characters in the principal block B0 of a finite group G. We prove that, if k(B0)−l(B0) = 1, then l(B0) ≥ p − 1 or else p = 11 and l(B0) = 9. This follows from a more general result that for every finite group G in which all non-trivial p-elements are conjugate, l(B0) ≥ p − 1 or else p = 11 and $$G/{{\bf{O}}_{{p^\prime }}}(G) \cong C_{11}^2\, \rtimes\,{\rm{SL}}(2,5)$$ G / O p ′ ( G ) ≅ C 11 2 ⋊ SL ( 2 , 5 ) . These results are useful in the study of principal blocks with few characters.We propose that, in every finite group G of order divisible by p, the number of irreducible Brauer characters in the principal p-block of G is always at least $$2\sqrt {p - 1} + 1 - {k_p}(G)$$ 2 p − 1 + 1 − k p ( G ) , where kp(G) is the number of conjugacy classes of p-elements of G. This indeed is a consequence of the celebrated Alperin weight conjecture and known results on bounding the number of p-regular classes in finite groups.

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“…and this indeed happens for all exceptional types and all d, except the single case of type E 7 and d = 4 (see also [HSF,p. 19]).…”

Section: Principal Blocks Of Almost Simple Groups Of Lie Typementioning

confidence: 66%

Bounding p-Brauer characters in finite groups with two conjugacy classes of p-elements

Hung,

Sambale,

Tiep

2024

Isr. J. Math.

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“…The next step came 41 years later, when J. Brandt proved in [Bra82] that k(B) = 2 if and only if |D| = 2. If B is the principal block, the classifications of defect groups for k(B) ≤ 6 have been obtained in [KS21], [RSV21] and [HSV23] but at the present time there is no classification for arbitrary blocks. The proof of Theorem A is achieved via a reduction to a problem on primitive permutation groups of low rank with a small number of projective characters, in the sense of Schur.…”

Section: Introductionmentioning

confidence: 89%

The blocks with four irreducible characters

Martínez¹,

Rizo²,

Sanus³

2022

Bulletin of London Math Soc

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“…So our project is to extend many and important results in the literature from the class of p-solvable groups to the class of p-constrained groups. In particular, according to recent work [29][30][31]. We raise the following questions:…”

Section: Future Workmentioning

confidence: 99%

On Height-Zero Characters in p-Constrained Groups

Algreagri,

Alghamdi

2023

Symmetry

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Consider G to be a finite group and p to be a prime divisor of the order |G| in the group G. The main aim of this paper is to prove that the outcome in a recent paper of A. Laradji is true in the case of a p-constrained group. We observe that the generalization of the concept of Navarro’s vertex for an irreducible character in a p-constrained group G is generally undefined. We illustrate this with a suitable example. Let ϕ∈Irr(G) have a positive height, and let there be an anchor group Aϕ. We prove that if the normalizer NG(Aϕ) is p-constrained, then Op´(NG(Aϕ))≠{1G}, where Op´(NG(Aϕ)) is the maximal normal p´ subgroup of NG(Aϕ). We use character theoretic methods. In particular, Clifford theory is the main tool used to accomplish the results.

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Height zero characters in principal blocks

Cited by 6 publications

References 57 publications

Bounding p-Brauer characters in finite groups with two conjugacy classes of p-elements

Bounding p-Brauer characters in finite groups with two conjugacy classes of p-elements

The blocks with four irreducible characters

On Height-Zero Characters in p-Constrained Groups

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