Groups in which the hypercentral factor group is a T-group.
The authors consider finite \(\text{T}_1\)-groups, i.e., the finite groups in which the factor group by its hypercenter is a group with transitivity of normality (T-groups). They describe some basic properties of such groups and highlight connections with its Frattini subgroup and its nilpotent radical. Here are some of the main results: Theorem A. Let \(G\) be a soluble finite \(\text{T}_1\)-group and \(R\) its nilpotent residual. Then 1. \(R\) is nilpotent of class at most 2. 2. \(G\) acts by conjugation on the factor-group of \(R\) by its derived subgroup as a group of power automorphisms. 3. The nilpotent residual of \(G^2\) is Abelian. Theorem D. Let \(G\) be a finite soluble group that is a product of two mutually permutable subgroups \(A\) and \(B\). If \(A\) is soluble and \(B\) is a \(\text{T}_1\)-group such that its nilpotent residual is \(G\)-invariant, then \(G\) is supersoluble. Corollary. The class of solvable \(\text{T}_1\)-groups is contained in the Fitting core of the formation of supersoluble groups.
scientific article; zbMATH DE number 5173067