Publ. Mat. 49 (2005), 351{362 THE LATTICE OF FITTING CLASSES WHICH ARE RIGHT EXTENSIBLE BY SOLUBLE GROUPS

In this paper we study the set of Fitting classes which are right extensible by soluble groups ordered by the inclusion relation. The consideration of the associated lattices gives rise to new Fitting classes and it allows to obtain some injectivity criteria for general Fitting classes. All considered groups are assumed to be nite. Set T = fT; T is a Fitting class such that T = TSg; where S is the class of soluble groups. The classes in T have a partic-ular interest in the Theory of Fitting classes of nite groups which are not necessarily soluble. For instance, in a previous paper the authors show that, if T 2 T, then each intermediate Fitting class F between T and hT; b(T)i is injective (that is each group has F-injectors) |in partic-ular, each Fitting class in Locksec(S) is injective, in accordance with a wider conjecture of Shemetkov [4, 11.117]|, being normal Fitting classes precisely those belonging to Locksec(hT; b(T)i). In this paper we study the set T ordered by the inclusion relation. The consideration of the lattices associated to T gives rise to new Fitting classes, as the complement in XN of the class of generalized nilpotent groups and the complement in TN of the class of nilpotent-constrained groups. The notations and terminology are referred to [1]. We denote by E the class of all ( nite) groups. Recall that a class of groups F is