scholarly journals A characterization of the group algebras of the finite groups

1966 ◽  
Vol 16 (2) ◽  
pp. 347-363 ◽  
Author(s):  
Marc Rieffel
Keyword(s):  
Finite Groups ◽  
Group Algebras

A study of enhanced power graphs of finite groups

2019 ◽  
Vol 19 (04) ◽  
pp. 2050062 ◽  
Author(s):  
Samir Zahirović ◽  
Ivica Bošnjak ◽  
Rozália Madarász
Keyword(s):  
Finite Groups ◽  
Prime Power ◽  
Cyclic Subgroup ◽  
Nilpotent Groups ◽  
Sufficient Condition ◽  
Finite Abelian Groups ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

The enhanced power graph [Formula: see text] of a group [Formula: see text] is the graph with vertex set [Formula: see text] such that two vertices [Formula: see text] and [Formula: see text] are adjacent if they are contained in the same cyclic subgroup. We prove that finite groups with isomorphic enhanced power graphs have isomorphic directed power graphs. We show that any isomorphism between undirected power graph of finite groups is an isomorphism between enhanced power graphs of these groups, and we find all finite groups [Formula: see text] for which [Formula: see text] is abelian, all finite groups [Formula: see text] with [Formula: see text] being prime power, and all finite groups [Formula: see text] with [Formula: see text] being square-free. Also, we describe enhanced power graphs of finite abelian groups. Finally, we give a characterization of finite nilpotent groups whose enhanced power graphs are perfect, and we present a sufficient condition for a finite group to have weakly perfect enhanced power graph.

scholarly journals Self-Dual Abelian Codes in Some Nonprincipal Ideal Group Algebras

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Parinyawat Choosuwan ◽  
Somphong Jitman ◽  
Patanee Udomkavanich
Keyword(s):  
Cyclic Codes ◽  
Group Algebras ◽  
Dual Codes ◽  
Complete Enumeration ◽  
Odd Order ◽  
Abelian Codes ◽  
Positive Integers ◽  
Ideal Group ◽  
Suitable Product

The main focus of this paper is the complete enumeration of self-dual abelian codes in nonprincipal ideal group algebrasF2k[A×Z2×Z2s]with respect to both the Euclidean and Hermitian inner products, wherekandsare positive integers andAis an abelian group of odd order. Based on the well-known characterization of Euclidean and Hermitian self-dual abelian codes, we show that such enumeration can be obtained in terms of a suitable product of the number of cyclic codes, the number of Euclidean self-dual cyclic codes, and the number of Hermitian self-dual cyclic codes of length2sover some Galois extensions of the ringF2k+uF2k, whereu2=0. Subsequently, general results on the characterization and enumeration of cyclic codes and self-dual codes of lengthpsoverFpk+uFpkare given. Combining these results, the complete enumeration of self-dual abelian codes inF2k[A×Z2×Z2s]is therefore obtained.

scholarly journals Thompson-like characterization of solubility for products of finite groups

2020 ◽  
Author(s):  
P. Hauck ◽  
L. S. Kazarin ◽  
A. Martínez-Pastor ◽  
M. D. Pérez-Ramos
Keyword(s):  
Finite Groups

LIE SOLVABLE GROUP ALGEBRAS OF DERIVED LENGTH THREE IN CHARACTERISTIC THREE

2012 ◽  
Vol 11 (05) ◽  
pp. 1250098 ◽  
Author(s):  
HARISH CHANDRA ◽  
MEENA SAHAI
Keyword(s):  
Solvable Group ◽  
Commutator Subgroup ◽  
Group Algebras ◽  
Derived Length ◽  
Engel Group

In this paper we provide a characterization of Lie solvable group algebras of derived length three over a field of characteristic three when G is a non-2-Engel group with abelian commutator subgroup.

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