Fixed point free action on groups of odd order.
Let \(G\) be a finite solvable group, let \(A\) be a nilpotent group of automorphisms of \(G\), and assume that \(C_G(A)=1\). It is conjectured that then the Fitting height \(h(G)\) of \(G\) is bounded above by the length \(\ell(A)\) of the longest chain of subgroups of \(A\). The authors prove this conjecture when \(G\) has odd order and \(A\) is Abelian of square-free exponent prime to \(6\). This improves, in this case, the linear bound proved by \textit{A. Turull} [in NATO ASI Ser., Ser. C, Math. Phys. Sci. 471, 377-400 (1995; Zbl 0847.20002)] to the best possible bound. Both the authors' and the reviewer's proofs follow the outline of the proof by \textit{E. C. Dade} [Ill. J. Math. 13, 449-514 (1969; Zbl 0195.04003)] together with the observation that for an Abelian group \(A\) of square free exponent acting fixed-point freely on a Fitting chain, one will be able to find a normal subgroup of \(A\) of prime order acting non-trivially on the top of the Fitting chain. More detailed analysis in the present paper yields the authors' stronger result.
scientific article; zbMATH DE number 5349061