Title:

Fitting and formation theory in locally finite groups

The theory of saturated formations introduced by Gaschütz 7 in 1963 is now an integral part of the study of finite soluble groups. Extensions of this theory have since been obtained by Stonehewer 18, for the class of periodic locally soluble groups with a normal locally nilpotent subgroup of finite index, and Tomlinson 21, for the class of periodic locally soluble FC groups. In the latter case of course conjugacy of the various types of subgroups concerned is replaced by local conjugacy. Wehrfritz 23 also developed a theory of basis normalizers and Carter subgroups for the class of all homomorphic images of periodic soluble linear groups. Much of this work was unified in 1971 in paper by Gardiner, Hartley and Tomkinson 6. They introduced a class U periodic locally soluble groups and showed that it is possible to obtain a theory of saturated formations in any subclass of U which is closed under taking subgroups and homomorphic images. Their work covers all the previous theories except that for periodic locally soluble FC  groups •. It is our aim to show that with a 'good' Sylow structure and a 'well behaved' group of automorphisms which permutes the Sylow structure transitively, a theory of saturated formations can always be constructed. The approach which we shall discuss gives a theory which covers all the previous theories including that for periodic locally soluble FC groups. This thesis is divided into two main parts and is organised as follows. In the next section we introduce our notation and terminology. In part one, which forms the bulk of the thesis, we shall define axiomatically what we mean by 'good' and 'well behaved', and in so doing introduce a class W of periodic locally soluble groups. We shall consider a fixed, but arbitrary, QS closed subclass K of W and a saturated K formation 7 satisfying certain conditions, and show that any Kgroup G possesses a unique A(G)  transitive class of 7projectors, where A(G) is a 'well behaved' group of automorphisms of G. In the final section of part one we extend the work of Chambers 4 to KA groups, and in particular we characterize the 7normalizers of KA groups by the covering/avoiding property. In part two, we introduce a class V of periodic locally soluble groups, where the automorphisms involved 'are the locally inner ones; this class properly contains the class of periodic locally soluble FC groups. We define what we mean by a Fitting class7 of Vgroups, and show that any Vgroup possesses a unique local .I conjugacy class of 7injectors. This extends the work of Tomkinson 22, and we follow his techniques when proving the above. The final sections of part two concern normal Fitting classes of Vgroups, where "re extend the work of Blessenohl and Gaschütz 2 and Lausch 15. ~ In particular we show that every non  trivial normal Fitting class of Vgroups admits one and only one normal Fitting pair ( up to isomorphic Fitting pairs ).
