Fixed point free action with some regular orbits
Suppose that a finite group \(A\) acts on a finite (solvable) group \(G\) so that \(C_G(A)=1\). Assume further that \((|A|,|G|)=1\). There is the long-standing conjecture that the Fitting height of \(G\) is bounded by the length of the longest chain of subgroups of \(A\). \textit{A. Turull} [J. Reine Angew. Math. 371, 67-91 (1986; Zbl 0587.20017)]\ proved the conjecture in the case where for every proper subgroup \(B\) of \(A\) and every \(B\)-invariant elementary Abelian section \(S\) of \(G\), there exists some \(v\in S\) such that \(C_B(v)=C_B(S)\) (we say that \(B\) has regular orbit on \(S\)). In the present paper, the author generalizes this theorem by proving that the conclusion still holds when we assume that regular orbits exist for certain crucial sections only. The proof follows the outline of the proof in [loc. cit.].
scientific article; zbMATH DE number 1066127