Normal Fitting classes were introduced by D. Blessenohl and W. Gaschütz [see Math. Z. 118, 1-8 (1970; Zbl 0208.03301)] in the soluble universe as Fitting classes whose injectors are normal in every soluble group. There exist different characterizations of normal Fitting classes in \(\mathcal S\) which reduce to considering different local normality concepts (such as \(A\)-normality, strong normality and quasinormality).
Let \({\mathcal F}\subseteq{\mathcal Y}\) be Fitting classes. (1) \(\mathcal F\) is said to be strongly normal in \(\mathcal Y\) if \({\mathcal F}\neq(1)\) and whenever \(G\in{\mathcal Y}\), \([G,\operatorname{Aut}(G)]\leq G_{\mathcal F}\) [see R. A. Bryce and J. Cossey, Math. Z. 141, 99-110 (1975; Zbl 0283.20016)].
The following two definitions were introduced by P. Hauck [Ph. D. thesis. Univ. Mainz (1977)].
(2) \(\mathcal F\) is said to be \(A\)-normal in \(\mathcal Y\) if \(G/G_{\mathcal F}\) is Abelian for all \(G\in{\mathcal Y}\).
(3) \(\mathcal F\) is said to be quasinormal in \(\mathcal Y\) if the following condition holds: if \(G\in{\mathcal F}\), \(p\) is a prime number and \(G\wr Z_p\in{\mathcal Y}\), then there exists a natural number \(m\geq 1\) such that \(G^m\wr Z_p\in{\mathcal F}\).
In this paper the relation between these concepts is studied in the finite universe. It is given an answer to a question proposed by K. Doerk and T. O. Hawkes [see p. 718 of the book Finite soluble groups (de Gruyter, 1992; Zbl 0753.20001)]. In particular it is obtained a sufficient condition implying the equivalence between \(A\)-normality and strong normality in a Fitting class \(\mathcal X\), which applies in the case when \({\mathcal X}=\mathcal{XS}\), and also if \(\mathcal X\) is the class \({\mathcal E}_\pi\) of all finite \(\pi\)-groups, for a set of primes \(\pi\).
Besides \(A\)-normality, with \(\mathcal F\)-dual pronormal subgroups playing the role of injectors, it is considered a natural generalization of normality corresponding to its original definition via injectors.
Let \(\mathcal F\) be a Fitting class. A subgroup \(U\) of a group \(G\) is said to be \(\mathcal F\)-dual pronormal in \(G\), and we write \(U {\mathcal F}\text{-dpn }G\) for short, if \(\langle U,U^g\rangle_{\mathcal F}\) is contained in \(U\), for each \(g\in G\) [see A. D’Aniello, Commun. Algebra 26, No. 2, 425-433 (1998; Zbl 0894.20024)].
The main results of this paper can be summarized as follows: If \({\mathcal Y}^*{\mathcal S}_p={\mathcal Y}^*\) for some prime \(p\), then the following statements are equivalent: (i) \(\mathcal F\) is \(A\)-normal in \(\mathcal Y\); (ii) whenever \(G_{\mathcal F}\subseteq H {\mathcal F}\text{-dpn }G\in{\mathcal Y}\), then \(H\triangleleft G\); (iii) whenever \(G_{\mathcal F}\subseteq H {\mathcal F}\text{-dpn }G\in{\mathcal Y}\), then \(H\neq G\); (iv) whenever \(G_{\mathcal F}\subseteq H {\mathcal F}\text{-dpn }G\in{\mathcal Y}\), then \(H\text{ pr }G\); (v) \(\mathcal F\) is strongly normal in \(\mathcal Y\); i.e., \({\mathcal F}^*={\mathcal Y}^*\). (See Theorem 3.17.)
If \({\mathcal Y}={\mathcal E}_\pi\) for some set of primes \(\pi\), then the above hypothesis holds, and so (i), (ii), (iii) are equivalent to (iv), (v) and also to (vi) whenever \(G_{\mathcal F}\subseteq H {\mathcal F}\text{-dpn }G\in{\mathcal Y}\), then \(H\text{ sn }G\). (See Theorem 3.22.)
Quasinormality turns out to be a weaker hypothesis (see Remark 3.20).