On Frobenius systems

Abstract

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.