Formation theoretic properties of certain locally finite groups

The theory of saturated formations introduced by Gaschütz in 1963 is now an integral part of the study of finite soluble groups. Extensions of this theory have since been obtained by Stonehewer (32) for the class of periodic locally soluble groups with a normal locally nilpotent subgroup of finite index, and Tomkinson (34), for the class of periodic locally soluble FC-groups. Wehrfritz (35) 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 a recent paper of Gardiner, Hartley and Tomkinson (7). They introduced a class U of 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, the situation there being somewhat different