Strong containment of fitting classes

Abstract

Let% and 9 be Fitting classes of finite soluble groups,% is said to be strongly contained in g if in each group G an X-injector of G is contained in a g-injector of G. In this case, we write % < g. Strong containment was first defined by Lockett [19], and has been studied by Makan [22] and by Bryce and Cossey [9]. Although it is immediate that .X < ?V only if E C g, the converse is by no means true [9, Introduction]. Let 9 be a Fitting class. The Fitting class g* is defined to be the class of groups G such that (G x G)p = (Gs x GF)(( g-l, g) 1 g E G). The intersection of all Fitting classes % such that %* = F* is denoted by F.+ , and is, of course, also a Fitting class [20]. In this paper, we consider a question based on one of Bryce and Cossey [9, 4.3b] concerning the relationship betweeh strong containment and the *-operations. Let % and g be Fitting classes such that 3 C g and 55 is strongly contained in ?V*.