Bounds on the fitting length of finite soluble groups with supersoluble Sylow normalisers

Abstract

We prove that, in a finite soluble group, all of whose Sylow normalisers are super-soluble, the Fitting length is at most 2 m + 2, where p m is the highest power of the smallest prime p dividing | G / G s | here G s is the supersoluble residual of G . The bound 2 m + 2 is best possible. However under certain structural constraints on G / G S , typical of the small examples one makes by way of experimentation, the bound is sharply reduced. More precisely let p be the smallest, and r the largest, prime dividing the order of a group G in the class under consideration. If a Sylow p –subgroup of G / G S acts faithfully on every r -chief factor of G / G S , then G has Fitting length at most 3.