Groups of even type which are not of even characteristic, I

Magaard, Kay; Stroth, Gernot

doi:10.1007/s11856-016-1313-x

Cited by 2 publications

(44 citation statements)

References 26 publications

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“…Now |Ω 1 (Z(O 2 (P 2 )))| = 4, as |Ω 1 (Z(S))| = 2 by [MaStr,Lemma 2.33]. Hence Ω 1 (Z(O 2 (P 2 ))) = Z 2 (S) by [MaStr,Lemma 2.35]. The assertion follows.…”

Section: Preliminariesmentioning

confidence: 88%

“…Hence there is some minimal parabolic P 2 ≤ P 1 such that P 2 /O 2 (P 2 ) ∼ = Σ 3 and P 2 ≤ C G (r). Now |Ω 1 (Z(O 2 (P 2 )))| = 4, as |Ω 1 (Z(S))| = 2 by [MaStr,Lemma 2.33]. Hence Ω 1 (Z(O 2 (P 2 ))) = Z 2 (S) by [MaStr,Lemma 2.35].…”

Section: Preliminariesmentioning

confidence: 96%

“…This is [GoLyS3,Lemma 4.10.5] and [GoLyS3,Table 5.3a] for M 11 . Lemma 2.6 ( [MaStr,Lemma 2.19]): Let L = L 4 (3), U 4 (3) or 2U 4 (3). Then the following holds:…”

Section: Preliminariesmentioning

confidence: 99%

“…Hypothesis 2.16 ( [MaStr,Hypothesis 2.27]): Let G = G(q), q = 2 n , be a simple group of Lie type, G ∼ = Sz(q), L 2 (q) or 2 F 4 (q) . Let R be a long root subgroup of G if G ∼ = Sp 2n (q), and a short root subgroup if G ∼ = Sp 2n (q).…”

Section: Preliminariesmentioning

confidence: 99%

“…Lemma 2.17 ( [MaStr,Lemma 2.28]): Assume Hypothesis 2.16 with G ∼ = L 3 (q), U 3 (q), Sp 4 (2) or G 2 (2) . Let L be a Levi complement in N G (R).…”

Section: Preliminariesmentioning

confidence: 99%

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Groups of even type which are not of even characteristic, II

Magaard

Stroth

2016

Isr. J. Math.

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“…Now |Ω 1 (Z(O 2 (P 2 )))| = 4, as |Ω 1 (Z(S))| = 2 by [MaStr,Lemma 2.33]. Hence Ω 1 (Z(O 2 (P 2 ))) = Z 2 (S) by [MaStr,Lemma 2.35]. The assertion follows.…”

Section: Preliminariesmentioning

confidence: 88%

Section: Preliminariesmentioning

confidence: 96%

“…This is [GoLyS3,Lemma 4.10.5] and [GoLyS3,Table 5.3a] for M 11 . Lemma 2.6 ( [MaStr,Lemma 2.19]): Let L = L 4 (3), U 4 (3) or 2U 4 (3). Then the following holds:…”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

“…Lemma 2.17 ( [MaStr,Lemma 2.28]): Assume Hypothesis 2.16 with G ∼ = L 3 (q), U 3 (q), Sp 4 (2) or G 2 (2) . Let L be a Levi complement in N G (R).…”

Section: Preliminariesmentioning

confidence: 99%

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Groups of even type which are not of even characteristic, II

Magaard

Stroth

2016

Isr. J. Math.

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Finite Groups Which are Almost Groups of Lie Type in Characteristic 𝐩

Parker,

Pientka,

Seidel

et al. 2023

Memoirs of the AMS

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Let p p be a prime. In this paper we investigate finite K { 2 , p } \mathcal K_{\{2,p\}} -groups G G which have a subgroup H ≤ G H \le G such that K ≤ H = N G ( K ) ≤ Aut ⁡ ( K ) K \le H = N_G(K) \le \operatorname {Aut}(K) for K K a simple group of Lie type in characteristic p p , and | G : H | |G:H| is coprime to p p . If G G is of local characteristic p p , then G G is called almost of Lie type in characteristic p p . Here G G is of local characteristic p p means that for all nontrivial p p -subgroups P P of G G , and Q Q the largest normal p p -subgroup in N G ( P ) N_G(P) we have the containment C G ( Q ) ≤ Q C_G(Q)\le Q . We determine details of the structure of groups which are almost of Lie type in characteristic p p . In particular, in the case that the rank of K K is at least 3 3 we prove that G = H G = H . If H H has rank 2 2 and K K is not PSL 3 ⁡ ( p ) \operatorname {PSL}_3(p) we determine all the examples where G ≠ H G \ne H . We further investigate the situation above in which G G is of parabolic characteristic p p . This is a weaker assumption than local characteristic p p . In this case, especially when p ∈ { 2 , 3 } p \in \{2,3\} , many more examples appear. In the appendices we compile a catalogue of results about the simple groups with proofs. These results may be of independent interest.

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Groups of even type which are not of even characteristic, I

Cited by 2 publications

References 26 publications

Groups of even type which are not of even characteristic, II

Groups of even type which are not of even characteristic, II

Finite Groups Which are Almost Groups of Lie Type in Characteristic 𝐩

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