The ϵJ-normalizers of a finite soluble group

Fitting classes and lattice formations I

“…The proposed required condition is motivated by Corollary 2.13. Some related results were obtained by Carter and Hawkes in [11] (see Theorem 2.14) and by Graddon in …”

Fitting classes and lattice formations II

“…Again the ^T-maximality of X in G implies that X = Tx-In particular, T < N. Therefore, CW/W is also an J?-projector terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700008880 [7] Fitting classes and lattice formations II 181…”

A Class of infinite soluble groups with an A-group condition

“…If £? is a saturated formation of finite soluble groups and G is a finite soluble group with abelian -residual G 8 , then G splits over G s and the complements are conjugate, being the $-normalizers of G [1]. This theorem has been extended to extensions of abelian Si-groups by hypercentral and hypercyclic groups in [3] and [10], where it is shown that if G is a group with hypercentral (hypercyclic) residual A such that G/A is hypercentral (hypercyclic) and A is an abelian ©i-group, then G splits over A and the complements are conjugate.…”

“…Thus M contains a free abelian subgroup X such that M/X is a Il-group for some finite set II of primes [5, Corollary 1 to Lemma 9.53]. If p^II, then M p " n X = X p " and so Let a&A; then X = (a G )isa finitely generated abelian normal subgroup of G and so there is an integer r such that X r is a free abelian normal subgroup of G. Let GIA = <f 1) ..., f k > with f f = t t A a p r element. Then X'XO/PO"'" is nilpotent.…”