# Bender's method

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In [group theory](/source/Group_theory), **Bender's method** is a method introduced by Bender (1970) for simplifying the local group theoretic analysis of the [odd order theorem](/source/Odd_order_theorem). Shortly afterwards he used it to simplify the [Walter theorem](/source/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](/source/Maximal_subgroup) *M* containing the [centralizer](/source/Centralizer) of an [involution](/source/Involution_(mathematics)#Group_theory), and its [generalized Fitting subgroup](/source/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](/source/P-group). 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, 110), and Kurzweil & Stellmacher (2004, Chapter 10.1).

## References

- Bender, Helmut (1970), ["On the uniqueness theorem"](http://projecteuclid.org/euclid.ijm/1256053074), *Illinois Journal of Mathematics*. **14** (3): 376–384, [doi:10.1215/ijm/1256053074](https://doi.org/10.1215/ijm/1256053074). [ISSN 0019-2082](https://www.worldcat.org/issn/0019-2082). MR 0262351
- Bender, Helmut (1970b), "On groups with abelian Sylow 2-subgroups", *[Mathematische Zeitschrift](/source/Mathematische_Zeitschrift)*. **117** (1–4): 164–176, [doi:10.1007/BF01109839](https://doi.org/10.1007/BF01109839). [ISSN 0025-5874](https://www.worldcat.org/issn/0025-5874). MR 0288180. [S2CID 120553015](https://api.semanticscholar.org/CorpusID:120553015)
- Bender, Helmut & Glauberman, George (1994), *Local analysis for the odd order theorem*, Vol. 188, London Mathematical Society Lecture Note Series, [Cambridge University Press](/source/Cambridge_University_Press), ISBN 978-0-521-45716-3. MR 1311244
- Gagen, Terence M. (1976), *Topics in finite groups*, [Cambridge University Press](/source/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*](https://www.ams.org/online_bks/surv402), Vol. 40, Mathematical Surveys and Monographs, Providence, R.I.: [American Mathematical Society](/source/American_Mathematical_Society), ISBN 978-0-8218-0390-5. MR 1358135
- Huppert, Bertram & Blackburn, Norman (1982), *Finite groups. III*, Vol. 243, Grundlehren der Mathematischen Wissenschaften, Berlin, New York: [Springer-Verlag](/source/Springer-Verlag), ISBN 978-3-540-10633-3. MR 662826
- Kurzweil, Hans & Stellmacher, Bernd (2004), *The theory of finite groups*, Universitext, Berlin, New York: [Springer-Verlag](/source/Springer-Verlag), ISBN 978-0-387-40510-0. MR 2014408

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