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A note on groups generated by involutions and sharply $ 2$-transitive groups


Authors: George Glauberman, Avinoam Mann and Yoav Segev
Journal: Proc. Amer. Math. Soc. 143 (2015), 1925-1932
MSC (2010): Primary 20B22; Secondary 20F99
DOI: https://doi.org/10.1090/S0002-9939-2014-12405-1
Published electronically: December 4, 2014
MathSciNet review: 3314102
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Abstract: Let $ G$ be a group generated by a set $ C$ of involutions which is closed under conjugation. Let $ \pi $ be a set of odd primes. Assume that either (1) $ G$ is solvable, or (2) $ G$ is a linear group.

We show that if the product of any two involutions in $ C$ is a $ \pi $-element, then $ G$ is solvable in both cases and $ G=O_{\pi }(G)\langle t\rangle $, where $ t\in C$.

If (2) holds and the product of any two involutions in $ C$ is a unipotent element, then $ G$ is solvable.

Finally we deduce that if $ \mathcal {G}$ is a sharply $ 2$-transitive (infinite) group of odd (permutational) characteristic, such that every $ 3$ involutions in $ \mathcal {G}$ generate a solvable or a linear group; or if $ \mathcal {G}$ is linear of (permutational) characteristic $ 0,$ then $ \mathcal {G}$ contains a regular normal abelian subgroup.


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Additional Information

George Glauberman
Affiliation: Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637
Email: gg@math.uchicago.edu

Avinoam Mann
Affiliation: Einstein Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel
Email: mann@math.huji.ac.il

Yoav Segev
Affiliation: Department of Mathematics, Ben-Gurion University, Beer-Sheva 84105, Israel
Email: yoavs@math.bgu.ac.il

DOI: https://doi.org/10.1090/S0002-9939-2014-12405-1
Received by editor(s): May 2, 2013
Received by editor(s) in revised form: October 20, 2013
Published electronically: December 4, 2014
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2014 American Mathematical Society