BOUNDS ON THE FITTING LENGTH OF FINITE SOLUBLE GROUPS WITH SUPERSOLUBLE SYLOW NORMALIZERS

We prove that, in a finite soluble group, all of whose Sylow normalisers are supersoluble, the Fitting length is at most 2m + 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 2m + 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.