On the unitary units of the group algebra 𝔽2kQ16

2015 ◽  
Vol 08 (01) ◽  
pp. 1550013 ◽  
Author(s):  
Zahid Raza ◽  
Riasat Ali
Keyword(s):  
Finite Field ◽  
Group Algebra ◽  
Quaternion Group ◽  
Characteristic 2

In this note, the structure of unitary unit groups V*(𝔽2kQ16) is investigated, where Q16, is quaternion group of order 16, and 𝔽2k is any finite field of characteristic 2, with 2k elements. In particular, we give the center Z(V*(𝔽2kQ16)) of unitary units subgroup V*(𝔽2kQ16) of group algebra 𝔽2kQ16. The structure of the unitary unit subgroup V*(𝔽2kQ16) is described with the help of the Z(V*(𝔽2kQ16)).

ON THE UNITARY UNITS OF THE GROUP ALGEBRA 𝔽2mM16

2013 ◽  
Vol 12 (08) ◽  
pp. 1350059 ◽  
Author(s):  
ZAHID RAZA ◽  
MAQSOOD AHMAD
Keyword(s):  
Finite Field ◽  
Group Algebra ◽  
Modular Group ◽  
Characteristic 2

In this note, we have given the center Z(V*(𝔽2mM16)) of unitary units subgroup V*(𝔽2mM16) of group algebra 𝔽2mM16, where M16 = 〈x, y | x8 = y2 = 1, xy = yx5〉 is the Modular group of order 16 and 𝔽2m is any finite field of characteristic 2, with 2m elements. The structure of the unitary unit subgroup V*(𝔽2mM16) of the group algebra 𝔽2mM16, is also described, see Theorem 3.1.

ON THE STRUCTURE OF THE UNITARY SUBGROUP OF THE GROUP ALGEBRA 𝔽2qD2n

2014 ◽  
Vol 13 (04) ◽  
pp. 1350139 ◽  
Author(s):  
ZAHID RAZA ◽  
MAQSOOD AHMAD
Keyword(s):  
Finite Field ◽  
Group Algebra ◽  
Dihedral Group ◽  
Characteristic 2 ◽  
Unitary Subgroup

We discuss the structure of the unitary subgroup V*(𝔽2qD2n) of the group algebra 𝔽2qD2n, where D2n = 〈x, y | x2n-1 = y2 = 1, xy = yx2n-1-1〉 is the dihedral group of order 2n and 𝔽2q is any finite field of characteristic 2, with 2q elements. We will prove that [Formula: see text], see Theorem 3.1.

Structure of the unitary subgroup of the group algebra 𝔽2n(QD)16

2018 ◽  
Vol 17 (04) ◽  
pp. 1850060
Author(s):  
Zahid Raza ◽  
Maqsood Ahmad
Keyword(s):  
Finite Field ◽  
Group Algebra ◽  
Abstract Algebra ◽  
Characteristic 2 ◽  
Unitary Subgroup

In this paper, we established the structure of unitary unit subgroup [Formula: see text] of the group algebra [Formula: see text], where [Formula: see text] is the Quasi-dihedral [D. S. Dummit and R. Foote, Abstract Algebra, 3rd edn. (Wiley, 2004), pp. 71–72] (Semi-Dihedral [B. Huppert, Endliche Gruppen (Springer, 1967), pp. 90–93]) group of order 16 and [Formula: see text] is any finite field of characteristic 2 with [Formula: see text] elements.

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].

UNITS OF THE GROUP ALGEBRA 𝔽2κ(C2 × D8)

2011 ◽  
Vol 10 (04) ◽  
pp. 643-647 ◽  
Author(s):  
JOE GILDEA
Keyword(s):  
Group Algebra ◽  
Unit Group ◽  
Cyclic Groups ◽  
Split Extensions ◽  
Characteristic 2

The structure of the unit group of the group algebra of the group C2 × D8 over any field of characteristic 2 is established in terms of split extensions of cyclic groups.

Maximal T-spaces of free associative algebras over a finite field

2018 ◽  
Vol 17 (04) ◽  
pp. 1850064
Author(s):  
C. Bekh-Ochir ◽  
S. A. Rankin
Keyword(s):  
Finite Field ◽  
Positive Characteristic ◽  
Associative Algebras ◽  
Prime Field ◽  
Characteristic 2 ◽  
Infinite Set

In earlier work, it was established that for any finite field [Formula: see text] and any nonempty set [Formula: see text], the free associative (nonunitary) [Formula: see text]-algebra on [Formula: see text], denoted by [Formula: see text], had infinitely many maximal [Formula: see text]-spaces, but exactly two maximal [Formula: see text]-ideals (each of which was shown to be a maximal [Formula: see text]-space). This raises the interesting question as to whether or not the maximal [Formula: see text]-spaces can be classified. However, aside from the two maximal [Formula: see text]-ideals, no examples of maximal [Formula: see text]-spaces of [Formula: see text] have been identified to this point. This paper presents, for each finite field [Formula: see text], an infinite set of proper [Formula: see text]-spaces [Formula: see text] of [Formula: see text], none of which is a [Formula: see text]-ideal. It is proven that for any distinct integers [Formula: see text], [Formula: see text]. Furthermore, it is proven that for the prime field [Formula: see text], [Formula: see text] any prime, [Formula: see text] is a maximal [Formula: see text]-space of [Formula: see text]. We conjecture that for any finite field [Formula: see text] of positive characteristic different from 2 and each integer [Formula: see text], [Formula: see text] is a maximal [Formula: see text]-space of [Formula: see text]. In characteristic 2, the situation is slightly different and we provide different candidates for maximal [Formula: see text]-spaces.

CONSTRUCTION OF SELF-DUAL RADICAL 2-CODES OF GIVEN DISTANCE

2012 ◽  
Vol 04 (04) ◽  
pp. 1250052
Author(s):  
CAROLIN HANNUSCH ◽  
PIROSKA LAKATOS
Keyword(s):  
Abelian Group ◽  
Group Algebra ◽  
Abelian Groups ◽  
Dual Code ◽  
Decomposition Formula ◽  
Construction Of Self ◽  
Characteristic 2

We prove that for arbitrary n ∈ ℕ and [Formula: see text] and for a field K of characteristic 2 there exists an abelian group G of order 2n such that one of the powers of the radical of the group algebra K[G] is a (2n, 2n-1, 2d)-self-dual code. These codes are constructed for abelian groups G with decomposition [Formula: see text] where a1 ≥ 3 and si ≥ 0(1 ≤ i ≤ 3).

scholarly journals The autocorrelation of the Möbius function and Chowla's conjecture for the rational function field in characteristic 2

2015 ◽  
Vol 373 (2040) ◽  
pp. 20140311 ◽  
Author(s):  
Dan Carmon
Keyword(s):  
Finite Field ◽  
Rational Function ◽  
Function Field ◽  
Möbius Function ◽  
Rational Function Field ◽  
Mobius Function ◽  
Characteristic 2

We prove a function field version of Chowla's conjecture on the autocorrelation of the Möbius function in the limit of a large finite field of characteristic 2, extending previous work in odd characteristic.

scholarly journals UNITS IN F2D2p

2013 ◽  
Vol 13 (02) ◽  
pp. 1350090 ◽  
Author(s):  
KULDEEP KAUR ◽  
MANJU KHAN
Keyword(s):  
Finite Field ◽  
Group Algebra ◽  
Dihedral Group ◽  
Unit Group ◽  
Normal Subgroups ◽  
Canonical Involution ◽  
Unitary Subgroup ◽  
Bicyclic Units

Let p be an odd prime, D2p be the dihedral group of order 2p, and F2 be the finite field with two elements. If * denotes the canonical involution of the group algebra F2D2p, then bicyclic units are unitary units. In this note, we investigate the structure of the group [Formula: see text], generated by the bicyclic units of the group algebra F2D2p. Further, we obtain the structure of the unit group [Formula: see text] and the unitary subgroup [Formula: see text], and we prove that both [Formula: see text] and [Formula: see text] are normal subgroups of [Formula: see text].

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