A note on groups generated by involutions and sharply $2$-transitive groups
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- by George Glauberman, Avinoam Mann and Yoav Segev PDF
- Proc. Amer. Math. Soc. 143 (2015), 1925-1932 Request permission
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
- MR Author ID: 267751
- 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
- MR Author ID: 225088
- Email: yoavs@math.bgu.ac.il
- 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
- © Copyright 2014 American Mathematical Society
- 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
- MathSciNet review: 3314102