CONDITIONS FOR A SCHUNCK CLASS TO BE A FORMATION

A Schunck class $\mathfrak{H}$ is determined by the class $\mathfrak{X}$ of primitives contained in $\mathfrak{H}$. We give necessary and sufficient conditions on $\mathfrak{X}$ for $\mathfrak{H}$ to be a saturated formation.

Schunck classes and projectors in a class of locally finite groups

We introduce a definition of a Schunck class of periodic abelian-by-finite soluble groups using major subgroups in place of the maximal subgroups used in Finite groups. This allows us to develop the theory as in the finite case proving the existence and conjugacy of projectors. Saturated formations are examples of Schunck classes and we are also able to obtain an infinite version of Gaschütz Ω-subgroups.

A Construction for Fitting-Schunck classes

A method for constructing Fitting-Schunck classes is given: the method is an adaptation of one given by C. L. Kanes for constructing Fitting formations, and generalizes the Fitting-Schunck class construction given by Cossey in 1981. A criterion for deciding which of the Fitting-Schunck classes so constructed are formations is given.

A Schunck class construction and a problem concerning primitive groups

Gaschütz has introduced the concept of a product of a Schunck class and a (saturated) formation (differing from the usual product of classes) and has shown that this product is a Schunck class provided that both of its factors consist of finite soluble groups. We investigate the same question in the context of arbitrary finite groups.

On Fitting classes that are Schunck classes

An example is given to show that a class of finite soluble groups that is both a Fitting class and a Schunck class need not be a formation. The novel feature of this class is that it is defined by imposing conditions on complemented chief factors of groups in it: this technique usually does not give rise to Fitting classes that are not formations.

Homomorphs and formations of given derived class

A homomorph H is a normal Schunck class if and only if there exists a derived class χ such that H = χ*; moreover, in this case one has H′ = χ (for the definitions, see below). These results give to the derived classes a decisive significance on the study of the normal Schunck classes (see (5)). The aim of this paper is to study the homomorphs H such that H′ is a fixed derived class: we prove that these homomorphs compose a complete and distributive lattice for the inclusion relation (the maximum of this lattice being a normal Schunck class). We construct the greatest and the smallest formations whose derived class is given. We prove finally that, except in trivial cases, a normal Schunck class is not a formation.

Closure operations for Schunck classes

In his Canberra lectures on finite soluble groups, [3], Gaschütz observed that a Schunck class (sometimes called a saturated homomorph) is {Q, Eφ, D0}-closed but not necessarily R0closed(*). In Problem 7·8 of the notes he then asks whether every {Q, Eφ, 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 Do < ro < Ro such that is a Schunck class if and only if = {QEφ, Ro}. 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 ⊆.