2-Weights for Unitary Groups

An, Jianbei

doi:10.2307/2154218

Cited by 11 publications

(44 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…From now on to the end of this paper, we always assume that p is a prime, q = p f with a positive integer f , and ℓ is a prime number different from p. We follow mainly the notation from [19], [8], [3], [4] and [5]. We first give some notation and conventions used throughout this paper.…”

Section: Some Notations and Conventions For Gl N (ǫQ)mentioning

confidence: 99%

“…We often abbreviateẑ for Res G N G (R) (ẑ). Now we recall the description of ℓ-weights of G in [2], [3], [4] and [5] and give some more notations and conventions.…”

Section: Weights Of Sl N (ǫQ)mentioning

confidence: 99%

“…We can make (5.2) because if for some z ∈ O ℓ ′ (Z), Res G Γ,δ,i C Γ,δ,i (ẑ) · θ Γ,δ,i = θ z.Γ,δ,i , then Res G Γ,δ,i N Γ,δ,i (ẑ) fixes every element of C Γ,δ,i . In fact, if ℓ = 2, then C Γ,δ,i has only one element by [3] and [4]. If ℓ is odd, and we assume that D Γ,δ,i = R m Γ ,α Γ ,γ,c , then by [2, p.14] and [5, p.10], N Γ,δ,i /D Γ,δ,i N m Γ ,α Γ ,γ /R m Γ ,α Γ ,γ × Y c /A c , for some subgroups Y c and A c .…”

Section: Weights Of Sl N (ǫQ)mentioning

confidence: 99%

“…Then by [2], [3], [4] and [5], the set of (n, ℓ)-admissible weight symbols is a labeling set for the Gconjugacy classes of ℓ-weights of G. We denote by (R w , ϕ w ) the ℓ-weight of G corresponding to the (n, ℓ)-admissible weight symbol w. The group O ℓ ′ (Z) acts on the set of (n, ℓ)-admissible weight symbols via z · [([σ 1 ], m 1 , λ (1) , K (1) ), . .…”

Section: Weights Of Sl N (ǫQ)mentioning

confidence: 99%

See 3 more Smart Citations

The blocks and weights of finite special linear and unitary groups

Feng

2019

Journal of Algebra

View full text Add to dashboard Cite

This paper has two main parts. First, we give a classification of the ℓ-blocks of finite special linear and unitary groups SL n (ǫq) in the non-defining characteristic ℓ ≥ 3. Second, we describe how the ℓ-weights of SL n (ǫq) can be obtained from the ℓ-weights of GL n (ǫq) when ℓ ∤ gcd(n, q − ǫ), and verify the Alperin weight conjecture for SL n (ǫq) under the condition ℓ ∤ gcd(n, q − ǫ). As a step to establish the Alperin weight conjecture for all finite groups, we prove the inductive blockwise Alperin weight condition for any unipotent ℓ-block of SL n (ǫq) if ℓ ∤ gcd(n, q − ǫ).

show abstract

Section: Some Notations and Conventions For Gl N (ǫQ)mentioning

confidence: 99%

“…We often abbreviateẑ for Res G N G (R) (ẑ). Now we recall the description of ℓ-weights of G in [2], [3], [4] and [5] and give some more notations and conventions.…”

Section: Weights Of Sl N (ǫQ)mentioning

confidence: 99%

Section: Weights Of Sl N (ǫQ)mentioning

confidence: 99%

Section: Weights Of Sl N (ǫQ)mentioning

confidence: 99%

See 2 more Smart Citations

The blocks and weights of finite special linear and unitary groups

Feng

2019

Journal of Algebra

View full text Add to dashboard Cite

show abstract

“…6.13,Thm. In [31], an approach to these cases was propounded based on the description of so-called radical r-subgroups and their normalizers in the symmetric and classical linear groups [38][39][40][41][42][43], and the case with 2, p ∈ π was taken up. A description of sporadic groups with the D π -property for arbitrary π was brought to a close in [29,Thms.…”

Section: Problemmentioning

confidence: 99%

The {ie210-01}-property in finite simple groups

Ревин

2008

Algebra Logic

View full text Add to dashboard Cite

Let π be some set of primes. A finite group is said to possess the D π -property if all of its maximal π-subgroups are conjugate. It is not hard to show that this property is equivalent to satisfaction of the complete analog of Sylow's theorem for Hall π-subgroups of a group. In the paper, we bring to a close an arithmetic description of finite simple groups with the D π -property, for any set π of primes. Previously, it was proved that a finite group possesses the D π -property iff each composition factor of the group has this property. Therefore, the results obtained mean in fact that the question of whether a given group enjoys the D π -property becomes purely arithmetic.Hall π-subgroup, then G is said to possess the D π -property [1]. A group with the E π -(C π -, D π -) property is also referred to as an E π -(C π -, D π -) group, respectively.Hall subgroups and groups with the D π -property were studied by many well-known authors (see, e.g., ). A classical result in this direction is a celebrated theorem by Hall, which was proved in [2,3], and independently, in [4]. The Hall theorem, we recall, says that a finite group is soluble iff it has the D π -property, for any set π of primes.Let π be fixed. Then the class of groups with the D π -property may turn out to be substantially wider than the class of all soluble groups, since, say, every π-or π -group possesses this property. Thus, it seems natural to pose the following: Problem 1. For every set π, describe all groups with the D π -property. The present paper brings to a close a series of works [28][29][30][31][32][33] devoted to this problem and contains a result which solves it out.The Hall theorem demonstrates that there exists a close connection between the composition structure of a finite group and the existence of the D π -property in that group, for all π. It turns out that this connection obtains even if the set π is fixed. It is not hard to show that every factor group of a group with the D π -property also has this property. Below are some problems which have been intensively studied in the mathematical literature starting in the 1950s (see [1, 5-13, 19, 20, 22]). Problem 2. Will an extension of a groupA by a group B possess the D π -property if A and B have this property? Problem 3. Will a normal subgroup of a group with the D π -property possess this property? In the general setting, Problem 2 was formulated in [13; 34, Question 3.62], and Problem 3 in [22; 34, Question 13.33]. Ph. Hall brought out a positive solution to Problem 2 for the case where a Hall π-subgroup is nilpotent in A and is soluble in B (see [1, Thm. D5]). Shemetkov in [10] solved this problem for the case where Sylow p-subgroups of A are cyclic for all p ∈ π, and obtained a series of other important results in this direction (see [11,12]). In [19] and [20], first attempts were made to use the theory of simple groups in dealing with Problem 2. Although a positive solution was obtained for several particular cases, the general approach to Problems 2 and 3 was outlined only in ...

show abstract