Groups whose subnormal subgroups are normal by-finite

Groups with many normal-by-finite subgroups

“…Here a subgroup H of a group G is called normal-by-finite if the core H G of H in G has finite index in H. Groups in which all subgroups are normal-by-finite have been considered in [3] and [10], where in particular it is proved that if G is a group with that property, and all periodic homomorphic images of G are locally finite, then G is abelian-byfinite. Moreover, the structure of groups in which every subnormal subgroup is normal-by-finite has been studied in [6]. Here we prove the following theorem.…”

Groups with All Subgroups Pronormal-by-Finite

“…Thus it is well known that F contains an abelian characteristic subgroup A of finite index. Moreover, there exists a G-invariant subgroup B of A such that A/B is periodic and G induces on B a group of power automorphisms (see [5], Theorem 2. whereD,Ē,L are normal subgroups ofḠ such thatḠ induces groups of power automorphisms on bothDL andĒL (see [5], Theorem 2.8). In particular,…”

“…As C K (F ) ≤ F and G /K is finite, it follows that G is a soluble group. Finally, Lemma 2.3 yields that all subnormal subgroups of G are normal-by-finite, and hence G is abelian-by-finite and G contains a metabelian subgroup of finite index (see [5], Theorem 3.4).…”

“…Moreover, Lemma 2.4 yields that G is abelian-by-finite, and hence it contains an abelian characteristic subgroup A of finite index. Since X/X G is finite for each subgroup X of A, there exists a subgroup B of A of finite index such that B = U × V , where all subgroups of U and all subgroups of V are normal in G (see [5], Theorem 2.8). Let x be any element of infinite order in the set G \ C G (U ).…”

“…In particular, X/X G is finite for each subgroup X of G . Thus G contains an abelian subgroup A of finite index such that A = U × V , where all subgroups of U and all subgroups of V are normal in G (see [5], Theorem 2.8). For each prime number p, let U p be the p-component of U .…”