Finite groups which are almost groups of Lie type in characteristic p / Chris Parker, Gerald Pientka, Andreas Seidel, Gernot Stroth.

Format:

Book

Author/Creator:

Parker, Christopher, 1961- author.

Pientka, Gerald, author.

Seidel, Andreas (Mathematician), author.

Stroth, Gernot, 1949- author.

Series:

Memoirs of the American Mathematical Society ; v. 1452.

Memoirs of the American Mathematical Society, 0065-9266 ; v. 1452.

Language:

English

Subjects (All):

Group theory.

Physical Description:

v, 182 pages ; 26 cm

Place of Publication:

Providence, RI : American Mathematical Society, [2023]

Summary:

"Let p be a prime. In this paper we investigate finite K{2,p}-groups G which have a subgroup H ≤ G such that K ≤ H = NG(K) ≤ Aut(K) for K a simple group of Lie type in characteristic p, and |G : H| is coprime to p. If G is of local characteristic p, then G is called almost of Lie type in characteristic p. Here G is of local characteristic p means that for all nontrivial p-subgroups P of G, and Q the largest normal p-subgroup in NG(P) we have the containment CG(Q) ≤ Q. We determine details of the structure of groups which are almost of Lie type in characteristic p. In particular, in the case that the rank of K is at least 3 we prove that G = H. If H has rank 2 and K is not PSL3(p) we determine all the examples where G = H. We further investigate the situation above in which G is of parabolic characteristic p. This is a weaker assumption than local characteristic p. In this case, especially when p ∈ {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." -- Provided by publisher

Contents:

Introduction

Preliminary group theoretical results

Identification theorems of some almost simple groups

Strongly p-embedded subgroups

Sylow embedded subgroups of linear groups

Main hypothesis an notation for the proof of the main theorms

The embedding of Q in G under hypothesis 6.2

The groups which satisfy hypothesis 6.2 with NF*(H)(Q) not soluble and NG(Q)≤/H

The groups with F*(H)≅PSL3(pe), p odd

The groups with F*(H)≅PSL3(2e) or Sp4(2e)'

The groups with F*(H)≅Sp2n(2e), n≥3

The groups with F*(H)≅F4(2e)

The case when p=2 and centralizer of some 2-central element of H is soluble

The groups with F*(H)≅G2(3e)

The groups with F*(H)≅PΩ+8(3) and NG(Q)≤/H

Proof of theorem 2 and theorem 3

Groups which satisfy hypothesis 6.2 with NG(Q)≤H and some p-local subgroup containing S not contained in H

Proof of theorem 4

Proof of theorem 1

Proof of main theorem 1 and main theorem 2

Properties of finite simple groups of lie type

Properties alternating groups

Small modules for finite simple groups

p-local properties of groups of lie type in characteristic p

Miscellanea

Notes:

"December 2023, volume 292, number 1452 (second of 6 numbers)."

Includes bibliographical references (pages 179-182).

ISBN:

1470467291

9781470467296

OCLC:

1416952336