Frattiniclasses and maximal subclasses, in particular of saturated formations of finite groups (Q1106941)

This paper collects some simple observations, taken from the second version of the author's Ph. D. thesis [Mainz, 1986; Zbl 0602.20022], concerning Frattini classes of classes of finite groups. For any closure operator A on classes of finite groups (in the sense of \textit{P. Hall} [Proc. Camb. Philos. Soc. 59, 531-553 (1963; Zbl 0118.036)]) the A- Frattini class \(\Phi_ A({\mathcal X})\) of an A-closed class \({\mathcal X}\) is the class of all \(G\in {\mathcal X}\) such that \({\mathcal X}=A{\mathcal Y}\) whenever \({\mathcal Y}\subseteq {\mathcal X}=A({\mathcal Y}\cup \{G\})\); for finitary A, it can be characterised as being the intersection of all maximal A-closed subclasses of \({\mathcal X}\), if such exist, and \({\mathcal X}\), otherwise. The non-technical result of the paper is the following straightforward Theorem: If \({\mathcal X}\) is a local formation of finite groups minimally locally defined by the formation function f, and if for all primes p in the characteristic of \({\mathcal X}\), f(p) is a local formation containing a non-trivial group, then \(\Phi_ A({\mathcal X})={\mathcal X}\), that is, \({\mathcal X}\) has no maximal local subformation.