scholarly journals A note onp-solvable and solvable finite groups

1994 ◽  
Vol 17 (4) ◽  
pp. 821-824
Author(s):  
R. Khazal ◽  
N. P. Mukherjee
Keyword(s):  
Finite Groups ◽  
Prime Divisor ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

The notion of normal index is utilized in proving necessary and sufficient conditions for a groupGto be respectively,p-solvable and solvable wherepis the largest prime divisor of|G|. These are used further in identifying the largest normalp-solvable and normal solvable subgroups, respectively, ofG.

CONTROL OF FUSION IN FUSION SYSTEMS

2006 ◽  
Vol 05 (06) ◽  
pp. 817-837 ◽  
Author(s):  
RADU STANCU
Keyword(s):  
Finite Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Fusion System ◽  
Fusion Systems ◽  
Necessary And Sufficient

Let P be a p-group and [Formula: see text] a fusion system on P. The aim of this paper is to give necessary and sufficient conditions on a subgroup Q of P for the normalizer [Formula: see text] to be [Formula: see text] itself. This generalizes a result of Gilotti and Serena on finite groups. As an application we find some classes of resistant p-groups, which are p-groups P such that the normalizer [Formula: see text] is equal to [Formula: see text], for any fusion system [Formula: see text] on P.

scholarly journals Trivial action on the tensor product of finite groups

1988 ◽  
Vol 30 (3) ◽  
pp. 271-274 ◽  
Author(s):  
R. J. Higgs
Keyword(s):  
Exact Sequence ◽  
Tensor Product ◽  
Direct Product ◽  
Semidirect Product ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Schur Multiplier ◽  
Necessary And Sufficient

Let G, H and K be finite groups such that K acts on both G and H. The action of K on G and H induces an action of K on their tensor product G ⊗ H, and we shall denote the K-stable subgroup of G ⊗ H by (G ⊗ H)K. In section 1 of this note we shall obtain necessary and sufficient conditions for (G ⊗ H)K = G ⊗ H. The importance of this result is that the direct product of G and H has Schur multiplier M(G × H) isomorphic to M(G) × M(H) × (G ⊗ H); moreover K: acts on M(G × H), and M(G × H)K is one of the terms contained in a fundamental exact sequence concerning the Schur multiplier of the semidirect product of K and G × H (see [3, (2.2.10) and (2.2.5)] for details). Indeed in section 2 we shall assume that G is abelian and use the fact that M(G) ≅ G ∧ G to find necessary and sufficient conditions for M(G)K = M(G).

scholarly journals Two remarks on amalgams of locally finite groups

1987 ◽  
Vol 36 (3) ◽  
pp. 461-468 ◽  
Author(s):  
Berthold J. Maier
Keyword(s):  
Finite Group ◽  
Finite Index ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Locally Finite Group ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
The Common ◽  
Necessary And Sufficient

We construct non amalgamation bases in the class of locally finite groups, and we present necessary and sufficient conditions for the embeddability of an amalgam into a locally finite group in the case that the common subgroup has finite index in both constituents.

scholarly journals Infinite groups admitting a faithful irreducible representation

2018 ◽  
Vol 17 (01) ◽  
pp. 1850005
Author(s):  
Fernando Szechtman ◽  
Anatolii Tushev
Keyword(s):  
Irreducible Representation ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Infinite Groups ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Necessary And Sufficient

Necessary and sufficient conditions for a group to possess a faithful irreducible representation are investigated. We consider locally finite groups and groups whose socle is essential.

On c#-normal subgroups of finite groups

2016 ◽  
Vol 16 (08) ◽  
pp. 1750160
Author(s):  
Guo Zhong ◽  
Shi-Xun Lin
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Normal Subgroups ◽  
A Finite Group ◽  
Necessary And Sufficient

Let [Formula: see text] be a subgroup of a finite group [Formula: see text]. We say that [Formula: see text] is a [Formula: see text]-normal subgroup of [Formula: see text] if there exists a normal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is a [Formula: see text]-subgroup of [Formula: see text]. In the present paper, we use [Formula: see text]-normality of subgroups to characterize the structure of finite groups, and establish some necessary and sufficient conditions for a finite group to be [Formula: see text]-supersolvable, [Formula: see text]-nilpotent and solvable. Our results extend and improve some recent ones.

scholarly journals On polygonal products of finitely generated abelian groups

1992 ◽  
Vol 45 (3) ◽  
pp. 453-462 ◽  
Author(s):  
Goansu Kim
Keyword(s):  
Finite Groups ◽  
Cyclic Subgroup ◽  
Abelian Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Residually Finite ◽  
Necessary And Sufficient

We prove that a polygonal product of polycyclic-by-finite groups amalgamating subgroups, with trivial intersections, is cyclic subgroup separable (hence, it is residually finite) if the amalgamated subgroups are contained in the centres of the vertex groups containing them. Hence a polygonal product of finitely generated abelian groups, amalgamating any subgroups with trivial intersections, is cyclic subgroup separable. Unlike this result, most polygonal products of four finitely generated abelian groups, with trivial intersections, are not subgroup separable (LERF). We find necessary and sufficient conditions for certain polygonal products of four groups to be subgroup separable.

BRAUER PAIRS OF VZ-GROUPS

2008 ◽  
Vol 07 (05) ◽  
pp. 663-670 ◽  
Author(s):  
ADRIANA NENCIU
Keyword(s):  
Finite Groups ◽  
Sufficient Conditions ◽  
Conjugacy Classes ◽  
Necessary And Sufficient Conditions ◽  
Irreducible Characters ◽  
Necessary And Sufficient ◽  
Brauer Pairs

Two non-isomorphic finite groups form a Brauer pair if there exist a bijection for the conjugacy classes and a bijection for the irreducible characters that preserve all the character values and the power map. A group is called a VZ-group if all its nonlinear irreducible characters vanish off the center. In this paper we give necessary and sufficient conditions for two non-isomorphic VZ-groups to form a Brauer pair.

Group Rings over Z(p) with FC Unit Groups

1980 ◽  
Vol 32 (5) ◽  
pp. 1266-1269 ◽  
Author(s):  
H. Merklen ◽  
C. Polcino Milies
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Group Ring ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Conjugacy Classes ◽  
Necessary And Sufficient Conditions ◽  
Group Rings ◽  
Necessary And Sufficient

Let RG denote the group ring of a group G over a commutative ring R with unity. We recall that a group is said to be an FC-group if all its conjugacy classes are finite.In [6], S. K. Sehgal and H. Zassenhaus gave necessary and sufficient conditions for U(RG) to be an FC-group when R is either Z, the ring of rational integers, or a field of characteristic 0.One of the authors considered this problem for group rings over infinite fields of characteristic p ≠ 2 in [5] and G. Cliffs and S. K. Sehgal [1] completed the study for arbitrary fields. Also, group rings of finite groups over commutative rings containing Z(p), a localization of Z over a prime ideal (p) were studied in [4].

scholarly journals Partial actions and cyclic kummer’s theory

2020 ◽  
pp. 2250032
Author(s):  
Víctor Marín ◽  
Andrés Cañas ◽  
Héctor Pinedo
Keyword(s):  
Cyclic Group ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Necessary And Sufficient Conditions ◽  
Galois Extensions ◽  
Radical Extension ◽  
Necessary And Sufficient

We introduce a theory of cyclic Kummer extensions of commutative rings for partial Galois extensions of finite groups, extending some of the well-known results of the theory of Kummer extensions of commutative rings developed by Borevich. In particular, we provide necessary and sufficient conditions to determine when a partial [Formula: see text]-Kummerian extension is equivalent to either a radical or an [Formula: see text]-radical extension, for some subgroup [Formula: see text] of the cyclic group [Formula: see text].

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