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December 2007 On Frobenius systems
Paul Lescot
Osaka J. Math. 44(4): 887-891 (December 2007).


By a Frobenius system on a finite group $G$, we mean the data, for each maximal solvable subgroup $M$ of $G$, of a normal subgroup $\mathcal{F}(M)$ of $M$, satisfying some of the properties of a Frobenius kernel, and subject to certain additional conditions. We prove that a finite group with a Frobenius system is either solvable (in which case we get a complete description), or isomorphic to $\mathit{SL}_{2}(K)$ (for $K$ a finite field of characteristic 2) or to a Suzuki group. The respective possibilities for the mapping $\mathcal{F}$ are then determined. This extends a previous result of ours (Nagoya Math. J. 165 (2002), 117--121) by removing the condition that each $M/\mathcal{F}(M)$ be abelian. Curiously enough, the Feit-Thompson Theorem is used in the proof.


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Paul Lescot. "On Frobenius systems." Osaka J. Math. 44 (4) 887 - 891, December 2007.


Published: December 2007
First available in Project Euclid: 7 January 2008

zbMATH: 1138.20022
MathSciNet: MR2383815

Primary: 20D60 , 20E99

Rights: Copyright © 2007 Osaka University and Osaka City University, Departments of Mathematics


Vol.44 • No. 4 • December 2007
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