Closure operations for Schunck classes

Abstract

In his Canberra lectures on finite soluble groups, [3], Gaschütz observed that a Schunck class (sometimes called a saturated homomorph) is {Q, E_φ, D₀}-closed but not necessarily R₀closed(*). In Problem 7·8 of the notes he then asks whether every {Q, E_φ, D₀}-closed class is a Schunck class. We show below with an example † that this is not the case, and then we construct a closure operation R₀ satisfying D_o < r_o < R_o such that is a Schunck class if and only if = {QEφ, R_o}. In what follows the class of finite soluble groups is universal. Let B denote the class of primitive groups. We recall that a Schunck class is one which satisfies: (a) = Q, and (b) contains all groups G such that Q(G) ∩ B ⊆.