Unit group of semisimple group algebras of some non-metabelian groups of order 120

2021 ◽  
pp. 2250059
Author(s):  
Gaurav Mittal ◽  
R. K. Sharma
Keyword(s):  
Symmetric Group ◽  
Unit Group ◽  
Semisimple Group ◽  
Group Algebras ◽  
Metabelian Groups

In this paper, we give the characterization of the unit groups of semisimple group algebras of some non-metabelian groups of order 120. This study completes the study of unit groups of semisimple group algebras of all groups up to order 120, except that of the symmetric group [Formula: see text] and groups of order 96.

A note on normal complement problem

2017 ◽  
Vol 16 (01) ◽  
pp. 1750011 ◽  
Author(s):  
K. Kaur ◽  
M. Khan ◽  
T. Chatterjee
Keyword(s):  
Group Algebra ◽  
Modular Group ◽  
Positive Characteristic ◽  
Unit Group ◽  
Semisimple Group ◽  
Group Algebras ◽  
Metacyclic Group ◽  
Infinite Class ◽  
Normal Complement ◽  
Characteristic 2

In this paper, we study the normal complement problem on semisimple group algebras and modular group algebras [Formula: see text] over a field [Formula: see text] of positive characteristic. We provide an infinite class of abelian groups [Formula: see text] and Galois fields [Formula: see text] that have normal complement in the unit group [Formula: see text] for semisimple group algebras [Formula: see text]. For metacyclic group [Formula: see text] of order [Formula: see text], where [Formula: see text] are distinct primes, we prove that [Formula: see text] does not have normal complement in [Formula: see text] for finite semisimple group algebra [Formula: see text]. Finally, we study the normal complement problem for modular group algebras over field of characteristic 2.

On *-clean non-commutative group rings

2016 ◽  
Vol 15 (08) ◽  
pp. 1650150 ◽  
Author(s):  
Hongdi Huang ◽  
Yuanlin Li ◽  
Gaohua Tang
Keyword(s):  
Finite Field ◽  
Local Ring ◽  
Group Algebra ◽  
Rational Numbers ◽  
Semisimple Group ◽  
Group Algebras ◽  
Group Rings ◽  
Dihedral Groups ◽  
Commutative Local Ring ◽  
Generalized Quaternion

A ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection (∗-invariant idempotent). In this paper, we consider the group algebras of the dihedral groups [Formula: see text], and the generalized quaternion groups [Formula: see text] with standard involution ∗. For the non-semisimple group algebra case, we characterize the ∗-cleanness of [Formula: see text] with a prime [Formula: see text], and [Formula: see text] with [Formula: see text], where [Formula: see text] is a commutative local ring. For the semisimple group algebra case, we investigate when [Formula: see text] is ∗-clean, where [Formula: see text] is the field of rational numbers [Formula: see text] or a finite field [Formula: see text] and [Formula: see text] or [Formula: see text].

scholarly journals Unit Groups of Semisimple Group Algebras of Abelianp-Groups over a Field

1997 ◽  
Vol 188 (2) ◽  
pp. 580-589 ◽  
Author(s):  
Nako A. Nachev ◽  
Todor Zh. Mollov
Keyword(s):  
Semisimple Group ◽  
Group Algebras

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.

The Hypercentre and the n-Centre of the Unit Group of an Integral Group Ring

1998 ◽  
Vol 50 (2) ◽  
pp. 401-411 ◽  
Author(s):  
Yuanlin Li
Keyword(s):  
Group Ring ◽  
Periodic Group ◽  
Unit Group ◽  
Complete Characterization ◽  
Integral Group Ring ◽  
Integral Group

AbstractIn this paper, we first show that the central height of the unit group of the integral group ring of a periodic group is at most 2. We then give a complete characterization of the n-centre of that unit group. The n-centre of the unit group is either the centre or the second centre (for n ≥ 2).

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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