One succinct version of Bender's method is the result that if M, N are two distinct maximal subgroups of a simple group with F*(M) ≤ N and F*(N) ≤ M, then there is a prime p such that both F*(M) and F*(N) are p-groups. This situation occurs whenever M and N are distinct maximal parabolic subgroups of a simple group of Lie type, and in this case p is the characteristic, but this has only been used to help identify groups of low Lie rank. These ideas are described in textbook form in Gagen (1976, p. 43), Huppert & Blackburn (1982, Chapter X. 15), Gorenstein, Lyons & Solomon (1996, p. 110, Chapter F.19), and Kurzweil & Stellmacher (2004, Chapter 10.1).
Referencesedit
Bender, Helmut (1970), "On the uniqueness theorem", Illinois Journal of Mathematics, 14 (3): 376–384, doi:10.1215/ijm/1256053074, ISSN 0019-2082, MR 0262351
Bender, Helmut (1970b), "On groups with abelian Sylow 2-subgroups", Mathematische Zeitschrift, 117 (1–4): 164–176, doi:10.1007/BF01109839, ISSN 0025-5874, MR 0288180, S2CID 120553015
Kurzweil, Hans; Stellmacher, Bernd (2004), The theory of finite groups, Universitext, Berlin, New York: Springer-Verlag, ISBN978-0-387-40510-0, MR 2014408
March 10, 2024
bender, method, group, theory, method, introduced, bender, 1970, simplifying, local, group, theoretic, analysis, order, theorem, shortly, afterwards, used, simplify, walter, theorem, groups, with, abelian, sylow, subgroups, bender, 1970b, gorenstein, walter, c. In group theory Bender s method is a method introduced by Bender 1970 for simplifying the local group theoretic analysis of the odd order theorem Shortly afterwards he used it to simplify the Walter theorem on groups with abelian Sylow 2 subgroups Bender 1970b and Gorenstein and Walter s classification of groups with dihedral Sylow 2 subgroups Bender s method involves studying a maximal subgroup M containing the centralizer of an involution and its generalized Fitting subgroup F M One succinct version of Bender s method is the result that if M N are two distinct maximal subgroups of a simple group with F M N and F N M then there is a prime p such that both F M and F N are p groups This situation occurs whenever M and N are distinct maximal parabolic subgroups of a simple group of Lie type and in this case p is the characteristic but this has only been used to help identify groups of low Lie rank These ideas are described in textbook form in Gagen 1976 p 43 Huppert amp Blackburn 1982 Chapter X 15 Gorenstein Lyons amp Solomon 1996 p 110 Chapter F 19 and Kurzweil amp Stellmacher 2004 Chapter 10 1 References editBender Helmut 1970 On the uniqueness theorem Illinois Journal of Mathematics 14 3 376 384 doi 10 1215 ijm 1256053074 ISSN 0019 2082 MR 0262351 Bender Helmut 1970b On groups with abelian Sylow 2 subgroups Mathematische Zeitschrift 117 1 4 164 176 doi 10 1007 BF01109839 ISSN 0025 5874 MR 0288180 S2CID 120553015 Bender Helmut Glauberman George 1994 Local analysis for the odd order theorem London Mathematical Society Lecture Note Series vol 188 Cambridge University Press ISBN 978 0 521 45716 3 MR 1311244 Gagen Terence M 1976 Topics in finite groups Cambridge University Press ISBN 978 0 521 21002 7 MR 0407127 Gorenstein D Lyons Richard Solomon Ronald 1996 The classification of the finite simple groups Number 2 Part I Chapter G Mathematical Surveys and Monographs vol 40 Providence R I American Mathematical Society ISBN 978 0 8218 0390 5 MR 1358135 Huppert Bertram Blackburn Norman 1982 Finite groups III Grundlehren der Mathematischen Wissenschaften vol 243 Berlin New York Springer Verlag ISBN 978 3 540 10633 3 MR 0662826 Kurzweil Hans Stellmacher Bernd 2004 The theory of finite groups Universitext Berlin New York Springer Verlag ISBN 978 0 387 40510 0 MR 2014408 Retrieved from https en wikipedia org w index php title Bender 27s method amp oldid 1169994738, wikipedia, wiki, book, books, library,
