Fitting classes and lattice formations I

Abstract

A lattice formation is a class of groups whose elements are the direct product of Hall subgroups corresponding to pairwise disjoint sets of primes. In this paper Fitting classes with stronger closure properties involving ^-subnormal subgroups, for a lattice formation & of full characteristic, are studied. For a subgroup-closed saturated formation #', a characterisation of the Sf-projectors of finite soluble groups is also obtained. It is inspired by the characterisation of the Carter subgroups as the <yK-projectors, «/f being the class of nilpotent groups.

Fitting classes and lattice formations II

“…The Fitting classes in this family are also saturated formations. Other examples of a different nature are also shown in [2,Examples I and II].…”

“…In a previous paper [2], ^"-Fitting classes associated to a lattice formation & containing jY, the class of all nilpotent groups, are introduced and studied. A lattice formation is a class of groups whose elements are the direct product of Hall subgroups corresponding to fixed pairwise disjoint sets of primes.…”

“…In [2,Theorem 3.9] a large family of ^"-Fitting classes, for every lattice formation & containing JV, is presented. The Fitting classes in this family are also saturated formations.…”

Carter and Gaschütz theories beyond soluble groups

“…Remark 4.12 It is not true in general that N π -projectors of π -soluble groups are exactly self-N π -Dnormalizing subgroups in N π . Otherwise, N π ∩ S would be either N or S, the class of all soluble groups, by [3,Proposition 4.1]. But we see next that a corresponding result to [3, Theorem 4.2] is still possible.…”

Injectors in $$\pi $$-Separable Groups

“…The class N π is a particular case of the so-called lattice formations, which are classes of groups whose elements are direct product of Hall subgroups corresponding to pairwise disjoint sets of primes. With the same flavour as N π -Fitting classes, though within the universe of finite soluble groups, L-Fitting classes, for general lattice formations L of soluble groups, were already defined in [3].…”