Classes of monoids with applications: formations of languages and multiply local formations of finite groups

Abstract

The Frattini subgroup $$\varPhi (G)$$ of a group G is the intersection of all maximal subgroups of G. A formation $${\mathfrak {F}}$$ of groups is said to be saturated if $$G/\varPhi (G) \in {\mathfrak {F}}$$ always implies $$G \in {\mathfrak {F}}$$ . A formation of finite groups is saturated iff it is local. A local satellite of $${\mathfrak {F}}$$ is a function with domain the set of all primes whose images are formations of finite groups. If the values of this function are themselves local formations, then this circumstance leads to the definition of multiply local formation. The languages corresponding to n-multiply local and totally local formations of finite groups are described.