Solubility of finite groups admitting a fixed-point-free automorphism of orderrst III

This paper, which is one of a series of four, contributes to the proof of the following Theorem.A finite group admitting a coprime fixed-point-free automorphism α of order rst (r, s andt distinct primes)is soluble. Here we prove that in a minimal counterexample to the above theorem the set ofα-invariant Sylowp-subgroupsP, such thatC _p(α ⁱ)≠1 for allα ⁱ≠1, generate a soluble subgroup.