Linear Groups of Small Degree over Fields of Finite Characteristic

Blau, Harvey I.; Zhang, J.P.

doi:10.1006/jabr.1993.1162

Cited by 6 publications

(16 citation statements)

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“…First we recall the description of absolutely irreducible non-solvable linear groups of degree less than p = char(k), relying on the main result of Blau and Zhang [5]:…”

Section: Linear Groups Of Low Degreementioning

confidence: 99%

“…(a) If p is a Fermat prime, then G is not p-solvable (equivalently, H is not solvable); (b) If p = 3, then H ∼ = SL 2 (3 a ) for all a ≥ 2; (c) If p = 5 and dim k W = 4, then H ∼ = Ω + 4 (5). Then dim k Ext 1 G (V, V ) ≤ 1 and dim k Ext 1 G (V, V * ) ≤ 1.…”

Section: Introductionmentioning

confidence: 99%

“…(ii) In the exceptional cases (p, dim k W, H) = (5,4, Ω + 4 (5)) or (p, H) = (3, SL 2 (3 a )) with a ≥ 2, Ext 1 G (V, V ) and Ext 1 G (V, V * ) are at most 2-dimensional.…”

Section: Introductionmentioning

confidence: 99%

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Adequate subgroups and indecomposable modules

Guralnick

Herzig

Tiep³

2017

J. Eur. Math. Soc.

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Abstract. The notion of adequate subgroups was introduced by Jack Thorne [59]. It is a weakening of the notion of big subgroups used by Wiles and Taylor in proving automorphy lifting theorems for certain Galois representations. Using this idea, Thorne was able to strengthen many automorphy lifting theorems. It was shown in [22] and [23] that if the dimension is smaller than the characteristic then almost all absolutely irreducible representations are adequate. We extend the results by considering all absolutely irreducible modules in characteristic p of dimension p. This relies on a modified definition of adequacy, provided by Thorne in [60], which allows p to divide the dimension of the module. We prove adequacy for almost all irreducible representations of SL2(p a ) in the natural characteristic and for finite groups of Lie type as long as the field of definition is sufficiently large. We also essentially classify indecomposable modules in characteristic p of dimension less than 2p − 2 and answer a question of Serre concerning complete reducibility of subgroups in classical groups of low dimension.

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“…First we recall the description of absolutely irreducible non-solvable linear groups of degree less than p = char(k), relying on the main result of Blau and Zhang [5]:…”

Section: Linear Groups Of Low Degreementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Adequate subgroups and indecomposable modules

Guralnick

Herzig

Tiep³

2017

J. Eur. Math. Soc.

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“…First we describe the structure of absolutely irreducible non-p-solvable linear groups of low degree, relying on the main result of Blau and Zhang [6]: (b) |P | = p, dim W = p − 1, and one of the following conditions holds: (b1) (H, p) = (SU n (q), (q n + 1)/(q + 1)), (Sp 2n (q), (q n + 1)/2), (2A 7 , 5), (3J 3 , 19), or (2Ru, 29).…”

Section: Linear Groups Of Low Degreementioning

confidence: 99%

“…This paper is organized as follows. In §2, based on results of [6], we describe the structure of (non-p-solvable) finite linear groups G < GL(V ) such that the dimension of irreducible G + -summands in V is less than p, cf. Theorem 2.4.…”

Section: Introductionmentioning

confidence: 99%

Adequate groups of low degree

Guralnick

Herzig

Tiep

2015

Algebra Number Theory

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Abstract. The notion of adequate subgroups was introduced by Jack Thorne [42]. It is a weakening of the notion of big subgroups used in generalizations of the Taylor-Wiles method for proving the automorphy of certain Galois representations. Using this idea, Thorne was able to strengthen many automorphy lifting theorems. It was shown in [22] that if the dimension is small compared to the characteristic then all absolutely irreducible representations are adequate. Here we extend the result by showing that, in almost all cases, absolutely irreducible kG-modules in characteristic p, whose irreducible G

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Some aspects of finite linear groups: A survey

Tiep¹,

Zalesskiǐ²

2000

J Math Sci

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Finite linear groups have been studied since the beginning of the century. The goal of this survey is to reflect recent results in this area, obtained mostly in the 90's and using the classification of fnite simple groups. It is a pleasure to thank the Institute for its hospitality. Linear Groups of Small DegreeLinear groups of degree <4 were classified at the beginning of the century, both for the zero and the prime characteristic cases, by using a kind of geometrical approach. New ideas, based on the representation theory elaborated by Brauer, Feit, and their successors, made it possible to classify linear groups of degree up

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Linear Groups of Small Degree over Fields of Finite Characteristic

Cited by 6 publications

References 0 publications

Adequate subgroups and indecomposable modules

Adequate subgroups and indecomposable modules

Adequate groups of low degree

Some aspects of finite linear groups: A survey

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