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

A. Ballester-Bolinches, J.C. Beidleman, R. Esteban-Romero, and M.F. Ragland

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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

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

First available in Project Euclid: 22 December 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $\pi$-length, ranks [See also 20F17] 20D20: Sylow subgroups, Sylow properties, $\pi$-groups, $\pi$-structure 20D35: Subnormal subgroups

Subnormal subgroup $\mathcal{T}$-group $\mathcal{PST}$-group finite solvable group


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.

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