�ber Vertauschbarkeit, normale Einbettung und Dominanz bei Fittingklassen endlicher aufl�sbarer Gruppen

Fitting pairs from direct limits and the lockett conjecture

“…Evidently cpg is the identity map for all G E J and, if G 4 H 4 K, then cp: cp$ = cp;. This means that {G/S(G); v $ ) G , H E E is a direct system of abelian groups, which ensures the existence of direct limit (A,9). Recall that this limit consists of a (possibly infinite) abelian group A and for each G E ( a homomorphism pG: G/S(G) -+ A such that &cpH = cpc whenever G 4 H .…”

“…Recall that, since X is normally embedded, the L,(X)-injectors of G have the form H R , where H is a Hall pl-subgroup of G and R is the L,(X)-radical of G (see [9]). …”

“…Then it is possible to construct a pair (A, cp), where A is an abelian group and cp assigns to each i E I a homomorphism cpi: A, -* A satisfying: 9) is universal with respect to these conditions. This construction is called the direct limit of the system.…”

On Local Embedding Properties of Injectors of Finite Soluble Groups

“…We recall the definition of a normally embedded subgroup and refer the reader to [4,1, Section 7] for further information about these subgroups. 6-normally embedded Fitting classes have been studied in detail by Lockett [8] and Doerk and Porta [5] (see [4,IX,Section 3]). By [4,IX,3.4(a)], each Fischer classthus in particular each subgroup-closed Fitting class-is an ©-normally embedded Fitting class, and according to [4,IX,2.9,3.7], the class 3 3 = (G | Soc 3 (G) < Z(G)) is a Lockett class which is not normally embedded in &.…”

“…For instance, Blessenohl and Gaschiitz [1], Hauck and Kienzle [7], Lockett [8] and Doerk and Porta [5] studied non-trivial Fitting classes whose injectors are respectively normal, (sub)modular, normally embedded and system permutable subgroups of G in each group G e @. These investigations can be generalized by considering non-trivial Fitting classes X and # of finite soluble groups such that X is contained in J and an X-injector of G satisfies a given embedding property e in G [3] Local embedding properties of injectors 25…”