Some remarks on maximal subgroups of finite
                    classical groups

Magaard, Kay

doi:10.1090/conm/694/13961

Cited by 7 publications

(9 citation statements)

References 32 publications

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“…Because of [15] we can also include PSL 2 subgroups in the list. (In the case of F 4 and p ≥ 5 this result was obtained by Magaard in [35], and for E 6 and H ∼ = PSL 2 (11), PSL 3 (3), PSU 3 (3) this result was obtained by Aschbacher in [1].) Theorem 1.2.…”

Section: Introductionmentioning

confidence: 63%

“…7 13 , 5 20 , 3 11 , 1 24 D5(a1) 7 28 , 3 7 , 2 8 , 1 15 A4 + A1 7 13 , 6 6 , 5 8 , 4 8 , 3 8 , 2 8 , 1 9 D5(a1) + A1 7 28 , 4 2 , 3 6 , 2 10 , 1 6 2A3 7 14 , 5 10 , 4 16 , 3 6 , 2 4 , 1 10 D4 + A2 7 28 , 5, 3 13 , 1 8 A4 + 2A1 7 15 , 6 4 , 5 8 , 4 8 , 3 9 , 2 8 , 1 4 D5(a1) + A2 7 28 , 5 3 , 4 2 , 3 6 , 2 4 , 1 3 A4 + A2 7 19 , 5 11 , 3 18 , 1 6 E6(a3) 7 28 , 5 7 , 3, 1 14 A4 + A2 + A1 7 19 , 6 2 , 5 7 , 4 8 , 3 7 , 2 6 , 1 3 E6(a3) + A1 7 28 , 6 2 , 5 3 , 4 2 , 3 2 , 2 4 , 1 3 A5 7 21 , 6 14 , 1 17 D6(a2) 7 29 , 5 4 , 4 4 , 3, 1 6 A4 + A3 7 24 , 6 2 , 5 3 , 4 6 , 3 6 , 2 4 , 1 3 E7(a5) 7 29 , 6 2 , 5, 4 4 , 3 3 , 1 3 A5 + A1 7 25 , 6 6 , 5 4 , 3, 2 4 , 1 6 E8(a7) 7 30 , 5 4 , 3 6 D4 7 28 , 1 52 A6 7 35 , 1 3 D4 + A1 7 28 , 3, 2 14 , 1 21 A6 + A1 7 35 , 3…”

Section: Class In Funclassified

“…Note that S 3 (M (E 6 )) G is 1-dimensional, and this yields a symmetric trilinear form on G, as is well known. Use of the trilinear form to study G and its subgroups was the focus of Aschbacher's series of papers [1, 2, 3, 4, 5], and was also used in Magaard's thesis on F 4 [35]. It also appears a little in work of Cohen-Wales [12] on Lie primitive subgroups of E 6 (C).…”

Section: Subgroups Of Ementioning

confidence: 99%

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On Medium-Rank Lie Primitive and Maximal Subgroups of Exceptional Groups of Lie Type

Craven

2023

Memoirs of the AMS

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We study embeddings of groups of Lie type H H in characteristic p p into exceptional algebraic groups G \mathbf {G} of the same characteristic. We exclude the case where H H is of type P S L 2 \mathrm {PSL}_2 . A subgroup of G \mathbf {G} is Lie primitive if it is not contained in any proper, positive-dimensional subgroup of G \mathbf {G} . With a few possible exceptions, we prove that there are no Lie primitive subgroups H H in G \mathbf {G} , with the conditions on H H and G \mathbf {G} given above. The exceptions are for H H one of P S L 3 ( 3 ) \mathrm {PSL}_3(3) , P S U 3 ( 3 ) \mathrm {PSU}_3(3) , P S L 3 ( 4 ) \mathrm {PSL}_3(4) , P S U 3 ( 4 ) \mathrm {PSU}_3(4) , P S U 3 ( 8 ) \mathrm {PSU}_3(8) , P S U 4 ( 2 ) \mathrm {PSU}_4(2) , P S p 4 ( 2 ) ′ \mathrm {PSp}_4(2)’ and 2 B 2 ( 8 ) {}^2\!B_2(8) , and G \mathbf {G} of type E 8 E_8 . No examples are known of such Lie primitive embeddings. We prove a slightly stronger result, including stability under automorphisms of G \mathbf {G} . This has the consequence that, with the same exceptions, any almost simple group with socle H H , that is maximal inside an almost simple exceptional group of Lie type F 4 F_4 , E 6 E_6 , 2 E 6 {}^2\!E_6 , E 7 E_7 and E 8 E_8 , is the fixed points under the Frobenius map of a corresponding maximal closed subgroup inside the algebraic group. The proof uses a combination of representation-theoretic, algebraic group-theoretic, and computational means.

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Section: Introductionmentioning

confidence: 63%

Section: Class In Funclassified

Section: Subgroups Of Ementioning

confidence: 99%

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On Medium-Rank Lie Primitive and Maximal Subgroups of Exceptional Groups of Lie Type

Craven

2023

Memoirs of the AMS

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“…As an example, we would like to mention (41) and (42) for the analysis of when a member of Class S can be contained in a member of Class C 2 . We refer the reader to (76) for a vision of what might be achieved.…”

Section: A Symplectic Formmentioning

confidence: 99%

Maximal subgroups of finite simple groups: classifications and applications

Roney-Dougal¹

2021

Surveys in Combinatorics 2021

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This paper surveys what is currently known about the maximal subgroups of the finite simple groups. After briefly introducing the groups themselves, if their maximal subgroups are completely determined then we present this classification. For the remaining finite simple groups our current knowledge is only partial: we describe the state of play, as well as giving some results that apply more generally. We also direct the reader towards computational resources for the construction of maximal subgroups.After this, we present three sample applications, selected because they combine group theoretical and combinatorial arguments, and because they use either or both of the detailed classifications and the looser statements that can be made about all maximal subgroups. In particular, we discuss results relating to generation, and the generating graph; results concerning bases; and some applications to computational complexity, in particular to graph colouring and other problems with no known polynomial-time solution.

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“…This is a special case of the general Irreducible Restriction Problem where we have an arbitrary almost quasi-simple group in place of A n . A major application of the Irreducible Restriction Problem is to the Aschbacher-Scott program on maximal subgroups of finite classical groups, see [1,6,13,22,26] for more details on this. For the purposes of the applications to the Aschbacher-Scott program we may assume that G is also almost quasi-simple, but we will not be making this additional assumption.…”

Section: Introductionmentioning

confidence: 99%

Irreducible restrictions of representations of alternating groups in small characteristics: Reduction theorems

Kleshchev¹,

Morotti²,

Tiep³

2020

Represent. Theory

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Building on reduction theorems and dimension bounds for symmetric groups obtained in our earlier work, we classify the irreducible restrictions of representations of the symmetric and alternating groups to proper subgroups. Such classification is known when the characteristic of the ground field is greater than 3, but the small characteristics cases require a substantially more delicate analysis and new ideas. Our results fit into the Aschbacher-Scott program on maximal subgroups of finite classical groups.

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Some remarks on maximal subgroups of finite classical groups

Cited by 7 publications

References 32 publications

On Medium-Rank Lie Primitive and Maximal Subgroups of Exceptional Groups of Lie Type

On Medium-Rank Lie Primitive and Maximal Subgroups of Exceptional Groups of Lie Type

Maximal subgroups of finite simple groups: classifications and applications

Irreducible restrictions of representations of alternating groups in small characteristics: Reduction theorems

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