## Publicacions Matemàtiques

### Some Local Properties Defining ${\mathcal T}_0$-Groups and Related Classes of Groups

#### Abstract

We call $G$ a $\operatorname{Hall}_{\mathcal X}$-group if there exists a normal nilpotent subgroup $N$ of $G$ for which $G/N'$ is an ${\mathcal X}$-group. We call $G$ a ${\mathcal T}_0$-group provided $G/\Phi(G)$ is a ${\mathcal T}$-group, that is, one in which normality is a transitive relation. We present several new local classes of groups which locally define $\operatorname{Hall}_{\mathcal X}$-groups and ${\mathcal T}_0$-groups where ${\mathcal X}\in\{ {\mathcal T},\mathcal {PT},\mathcal {PST}\}$; the classes $\mathcal {PT}$ and $\mathcal {PST}$ denote, respectively, the classes of groups in which permutability and S-permutability are transitive relations.

#### Article information

Source
Publ. Mat., Volume 60, Number 1 (2016), 265-272.

Dates
First available in Project Euclid: 22 December 2015

Ballester-Bolinches, A.; Beidleman, J.C.; Esteban-Romero, R.; Ragland, M.F. Some Local Properties Defining ${\mathcal T}_0$-Groups and Related Classes of Groups. Publ. Mat. 60 (2016), no. 1, 265--272. https://projecteuclid.org/euclid.pm/1450818490