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.

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 On infinite groups in which all abelian subgroups are locally cyclic

2021 ◽  
Author(s):  
Costantino Delizia ◽  
Chiara Nicotera
Keyword(s):  
Finite Groups ◽  
Abelian Subgroup ◽  
Periodic Case ◽  
Infinite Groups ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Periodic Groups ◽  
Abelian Subgroups

AbstractThe structure of locally soluble periodic groups in which every abelian subgroup is locally cyclic was described over 20 years ago. We complete the aforementioned characterization by dealing with the non-periodic case. We also describe the structure of locally finite groups in which all abelian subgroups are locally cyclic.

scholarly journals The Structure of Locally Finite Two-Connected Graphs

10.37236/1211 ◽  
1995 ◽  
Vol 2 (1) ◽  
Author(s):  
Carl Droms ◽  
Brigitte Servatius ◽  
Herman Servatius
Keyword(s):  
Connected Graph ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Locally Finite ◽  
Connected Graphs ◽  
Finite Graphs ◽  
Locally Finite Graphs ◽  
Necessary And Sufficient ◽  
Block Tree ◽  
Labeled Tree

We expand on Tutte's theory of $3$-blocks for $2$-connected graphs, generalizing it to apply to infinite, locally finite graphs, and giving necessary and sufficient conditions for a labeled tree to be the $3$-block tree of a $2$-connected graph.

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).

On the structure of groups whose non-abelian subgroups are subnormal

2014 ◽  
Vol 12 (12) ◽  
Author(s):  
Leonid Kurdachenko ◽  
Sevgi Atlıhan ◽  
Nikolaj Semko
Keyword(s):  
Finite Groups ◽  
Infinite Groups ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Abelian Subgroups

AbstractThe main aim of this article is to examine infinite groups whose non-abelian subgroups are subnormal. In this sense we obtain here description of such locally finite groups and, as a consequence we show several results related to such groups.

Matchings in Countable Graphs

1977 ◽  
Vol 29 (1) ◽  
pp. 165-168 ◽  
Author(s):  
K. Steffens
Keyword(s):  
Perfect Matching ◽  
Sufficient Conditions ◽  
Finite Graph ◽  
Necessary And Sufficient Conditions ◽  
Locally Finite ◽  
Finite Graphs ◽  
Locally Finite Graphs ◽  
Arbitrary Graphs ◽  
Necessary And Sufficient

Tutte [9] has given necessary and sufficient conditions for a finite graph to have a perfect matching. Different proofs are given by Brualdi [1] and Gallai [2; 3]. The shortest proof of Tutte's theorem is due to Lovasz [5]. In another paper [10] Tutte extended his conditions for a perfect matching to locally finite graphs. In [4] Kaluza gave a condition on arbitrary graphs which is entirely different from Tutte's.

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.

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.

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