Normal -Fitting classes

Abstract

The authors together with M. J. Karbe [///. J. Math. 33 (1989) have considered Fitting classes X of S,-groups and, under some rather strong restrictions, obtained an existence and conjugacy theorem for 3£-injectors. Results of Menegazzo and Newell show that these restrictions are, in fact, necessary.The Fitting class X is normal if, for each G e S , , G$ is the unique 3E-injector of G. X is abelian normal if, for each GeS 1 ( GjSG'. For finite soluble groups these two concepts coincide but the class of Cernikov-bynilpotent S,-groups is an example of a nonabelian normal Fitting class of S r groups. In all known examples in which I-injectors exist X is closely associated with some normal Fitting class (the Cernikov-by-nilpotent groups arise from studying the locally nilpotent injectors).Here we investigate normal Fitting classes further, paying particular attention to the distinctions between abelian and nonabelian normal Fitting classes. Products and intersections with (abelian) normal Fitting classes lead to further examples of Fitting classes satisfying the conditions of the existence and conjugacy theorem.

Wreath products and Fitting classes of $\mathfrak S_1$-groups

“…. We show that X* = F and then it follows from Theorem 2.1 of [4] that X is abelian normal . Suppose then that X* :~.IZ and let G E fi \ X* .…”

“…Since we are only considering Cernikov groups wc can choose a counterexample (G, p) such that G is minimal ; that is, the lemma holds f'or all proper subgroups of G . By Lemma 3.1 of [4], :X contains all hypercentral (_ 11 -groups . In particular Z E Y and so G :~1 .…”

“…Let G E .A ; then G/G-T is abelian . By Lemma 3.1 of [4], sj n .13 C X and so G/G~is finitely generated [1, Theorem 3.1] . We prove by induction on the torsion-free rank r of G/G~that G E sX.…”

“…In [3], [4] it was observed that in all the known examples, the subgroup M above can be chosen to be the 2j-radical of G for some normal .A-Fitting class 1-j . A A-Fitting class 2,) is normal if G-T is the unique 2j-injector of G for each G E A.…”

“…For example, let nj be an abelian normal C 1 -Fitting class such that there is an C1-group G with G/G,ĩ nfinito cyclic . Such an C 1-Fitting class 1 -9 and C1-group G are constructed in [4] . Consider :X = H2 n nj ; 3E* = 512 and so :X is not abelian normal and :X is not a Lockett class.…”