On finite T-groups and the Wielandt subgroup

Abstract

Abstract. The Wielandt subgroup of a group G is the intersection of the normalizers of all the subnormal subgroups of G. A T-group is a group in which all the subnormal subgroups are normal, or, equivalently, a group coinciding with its Wielandt subgroup. We investigate the Wielandt subgroup of finite solvable groups and, in particular, find new properties and characterizations (see Theorems 1, 2 and Corollaries 4, 6) for this subgroup in the case that G is metanilpotent. Furthermore, we provide new characterizations for finite solvable T-groups in Theorem 7.

On T-groups, supersolvable groups, and maximal subgroups

“…In particular, such groups are supersolvable and each subgroup of a solvable T -group is itself a T -group (the latter property is far from being obvious). T -groups have been widely studied, and characterizations of solvable T -groups are given, for instance, in [1][2][3]12,14] and [10].…”

Finite groups in which normality, permutability or Sylow permutability is transitive

“…The motivation for these results comes from results of Bryce, Cossey, BallesterBolinches, Esteban-Romero and Kaplan [10,4,13]. We obtain a characterization of C p -groups (Theorem 3.6) like Kaplan's characterization of soluble T -groups [13].…”

“…In [13] Kaplan showed that G is a soluble T -group if and only if ω * (G) = G. A p-soluble group G has plength one (l p (G) = 1) if and only if for every prime p, G/O p (G) has a normal Sylow p-subgroup (see [19]). Kaplan also proved that if G is a metanilpotent group, then ω(G) = ω * (G).…”

On Generalised Fc-Groups in Which Normality Is a transitive Relation

“…In their seminal papers, G. Zacher [22] (1952) and W. Gaschütz [7] (1957) described the structure of finite soluble T-groups. Later, many authors studied finite groups in which normality is transitive, and more characterisations were obtained (see for instance [1,2,3,9,10,14]) in terms of properties of subgroups. It turns out that every finite soluble T-group is aT-group (that is, a group in which every subgroup is a T-group).…”