Fitting functors and radicals of finite groups. (Q662642)

If \(\mathfrak X\) is a class of groups, then a map \(f\) that assigns to each group \(G\) of the class \(\mathfrak X\) a nonempty system of its subgroups \(f(G)\) is said to be a subgroup \(\mathfrak X\)-functor if \(\alpha(f(G))=f(\alpha(G))\) for every isomorphism \(\alpha\) of \(G\). For a nonempty class \(\mathfrak X\), a subgroup \(\mathfrak X\)-functor \(f\) is called Fitting or radical if \(f(X)=\{X\cap H\mid H\in f(G)\}\) for every \(\mathfrak X\)-group \(G\) and every normal \(\mathfrak X\)-subgroup \(X\). In this paper, the authors develop methods for recognizing the Fitting classes and radicals of finite groups by Fitting functors and prescribed properties of Hall \(\pi\)-subgroups. The first goal is to discover some general laws for generating Fitting classes by means of Fitting \(\mathfrak X\)-functors. The main result of the paper is some description of the radicals of \(\pi\)-soluble groups in terms of Fitting \(\mathfrak X\)-functors. Some applications of this result are also given.