An $ \mathfrak{X} $ -crown of a finite soluble group

Abstract

Let G be a finite soluble group and $ {\Phi_\mathfrak{X}}(G) $ an intersection of all those maximal subgroups M of G for which $ {{G} \left/ {{{\text{Cor}}{{\text{e}}_G}(M)}} \right.} \in \mathfrak{X} $ . We look at properties of a section $ F\left( {{{G} \left/ {{{\Phi_\mathfrak{X}}(G)}} \right.}} \right) $ , which is definable for any class $ \mathfrak{X} $ of primitive groups and is called an $ \mathfrak{X} $ -crown of a group G. Of particular importance is the case where all groups in $ \mathfrak{X} $ have equal socle length.