Bender's method

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title: "Bender's method" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["group-theory"] description: "Group theory method by Bender" topic_path: "general/group-theory" source: "https://en.wikipedia.org/wiki/Bender's_method" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Group theory method by Bender ::

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*(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

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