Search results for "Finite Solvable Group."
showing 4 items of 4 documents
Injectors with a normal complement in a finite solvable group
2011
Abstract Suppose G is a finite solvable group, and H is a subgroup with a normal complement in G. We shall find necessary and sufficient conditions (some of which are related to the properties of coprime actions) for H to be an injector in G. We shall also use these criteria to find characterizations of injectors which need not have a normal complement.
Injectors with a central socle in a finite solvable group
2013
Abstract In response to an Open Question of Doerk and Hawkes (1992) [2, IX §4, p. 628] , we shall describe three constructions for the Z π -injectors of a finite solvable group, where Z π is the Fitting class formed by the finite solvable groups whose π -socle is central (and π is a set of prime numbers).
Some local properties defining $\mathcal T_0$-groups and related classes of groups
2016
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.
Some local properties defining $T_0$-groups and related classes of groups
2016
[EN] We call G a Hall_X -group if there exists a normal nilpotent subgroup N of G for which G/N' is an X -group. We call G a T0 -group provided G/\Phi(G) is a T -group, that is, one in which normality is a transitive relation. We present several new local classes of groups which locally define Hall_X -groups and T_0 -groups where X ∈ {T , PT , PST }; the classes PT and PST denote, respectively, the classes of groups in which permutability and S-permutability are transitive relations.