On dualization of a result of Bryce and Cossey theory

Let σ be a partition of the set of all primes P . Let G be a finite group and F be a Fitting class of finite groups. In the theory of formations of finite soluble groups, a well known result of Bryce and Cossey is: a local formation F is a Fitting class if and only if every value of the canonical formation function F of F is a Fitting class. In this paper, we give the dual theory of the result of Bryce and Cossey. We proved that an σ -local Fitting class F is a formation if and only if every value of the canonical σ -local H σ -function of F is a formation.