Bender's method
{{Short description|Group theory method by Bender}}
In group theory, Bender's method is a method introduced by {{harvtxt|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 {{harvtxt|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 {{harvtxt|Gagen|1976|p=43}},
{{harvtxt|Huppert|Blackburn|1982|loc=Chapter X. 15}}, {{harvtxt|Gorenstein|Lyons|Solomon|1996|loc=Chapter F.19|p=110}}, and {{harvtxt|Kurzweil|Stellmacher|2004|loc=Chapter 10.1}}.
References
- {{Citation | last1=Bender | first1=Helmut | title=On the uniqueness theorem | url=http://projecteuclid.org/euclid.ijm/1256053074 | mr=0262351 | year=1970 | journal=Illinois Journal of Mathematics | issn=0019-2082 | volume=14 | issue=3 | pages=376–384| doi=10.1215/ijm/1256053074 | doi-access=free }}
- {{Citation | last1=Bender | first1=Helmut | title=On groups with abelian Sylow 2-subgroups | doi=10.1007/BF01109839 | mr=0288180 | year=1970b | journal=Mathematische Zeitschrift | issn=0025-5874 | volume=117 | issue=1–4 | pages=164–176| s2cid=120553015 }}
- {{Citation | last1=Bender | first1=Helmut | last2=Glauberman | first2=George | author2-link=George Glauberman | title=Local analysis for the odd order theorem | publisher=Cambridge University Press | series=London Mathematical Society Lecture Note Series | isbn=978-0-521-45716-3 | mr=1311244 | year=1994 | volume=188}}
- {{Citation | last1=Gagen | first1=Terence M. | title=Topics in finite groups | publisher=Cambridge University Press | isbn=978-0-521-21002-7 | mr=0407127 | year=1976}}
- {{Citation | last1=Gorenstein | first1=D. | author1-link=Daniel Gorenstein | last2=Lyons | first2=Richard | last3=Solomon | first3=Ronald | title=The classification of the finite simple groups. Number 2. Part I. Chapter G | url=https://www.ams.org/online_bks/surv402 | publisher=American Mathematical Society | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-0390-5 | mr=1358135 | year=1996 | volume=40}}
- {{Citation | last1=Huppert | first1=Bertram | author1-link=Bertram Huppert | last2=Blackburn | first2=Norman | title=Finite groups. III | publisher=Springer-Verlag | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften | isbn=978-3-540-10633-3 | mr=662826 | year=1982 | volume=243}}
- {{Citation | last1=Kurzweil | first1=Hans | last2=Stellmacher | first2=Bernd | title=The theory of finite groups | publisher=Springer-Verlag | location=Berlin, New York | series=Universitext | isbn=978-0-387-40510-0 | mr=2014408 | year=2004}}