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group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, Bender's method is a method introduced by 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 , 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 In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', ...
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 , , , and .


References

* * * * * * * {{Citation , last1=Kurzweil , first1=Hans , last2=Stellmacher , first2=Bernd , title=The theory of finite groups , publisher=
Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 ...
, location=Berlin, New York , series=Universitext , isbn=978-0-387-40510-0 , mr=2014408 , year=2004 Group theory