Generic fixed point free action of arbitrary finite groups (Q1063687)

Let A be a finite group acting on the solvable group G with the orders of A and G relatively prime and \(C_ G(A)\) trivial. If the action of A on G is ''generic'', a sharp bound is obtained for the Fitting height of G. The action of A on G is generic if for the action of all proper subgroups of A on sections of G there are ''enough regular orbits and modules''. It is known that if A is solvable then the Fitting height of G is bounded by the number of primes, counting multiplicities, that divide the order of A in certain special cases and in general by twice that number. The sharp bound obtained here for the case of an arbitrary finite group A whose action on G is generic is l(A), the largest integer n such that there exists a sequence \(1=S_ 0\subset S_ 1\subset S_ 2\subset...\subset S_ n=A\) of subgroups of A.