Bender's method
In group theory, 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 of an involution, and its generalized Fitting subgroup F*.
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* ≤ N and F* ≤ M, then there is a prime p such that both F* and F* 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,
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