Unitary units of the group algebra of modular groups

2020 ◽  
pp. 2250027
Author(s):  
Zsolt Adam Balogh
Keyword(s):  
Finite Field ◽  
Group Algebra ◽  
Modular Group ◽  
Modular Groups ◽  
Characteristic Two ◽  
Unitary Subgroup

Let [Formula: see text] be the group algebra of the modular group [Formula: see text] over a finite field [Formula: see text] of characteristic two. We calculate the order of the ∗-unitary subgroup of the group algebra [Formula: see text] and describe the structure of the ∗-unitary subgroup in the case when [Formula: see text].

On the order of unitary subgroup of the modular group algebra 𝔽2kD2N

2015 ◽  
Vol 14 (08) ◽  
pp. 1550129 ◽  
Author(s):  
Neha Makhijani ◽  
R. K. Sharma ◽  
J. B. Srivastava
Keyword(s):  
Group Algebra ◽  
Modular Group ◽  
Dihedral Group ◽  
Group Of Units ◽  
Modular Group Algebra ◽  
Canonical Involution ◽  
Unitary Subgroup

Let 𝔽qD2N be the group algebra of D2N, the dihedral group of order 2N, over 𝔽q = GF (q). In this paper, we compute the order of the unitary subgroup of the group of units of 𝔽2kD2N with respect to the canonical involution ∗.

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].

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.

Units of commutative group rings over polynomial ring

2018 ◽  
Vol 13 (01) ◽  
pp. 2050021
Author(s):  
S. Kaur ◽  
M. Khan
Keyword(s):  
Abelian Group ◽  
Finite Field ◽  
Polynomial Ring ◽  
Group Algebra ◽  
Modular Group ◽  
Finite Abelian Group ◽  
Unit Group ◽  
Group Rings ◽  
Modular Group Algebra ◽  
Univariate Polynomial

In this paper, we obtain the structure of the normalized unit group [Formula: see text] of the modular group algebra [Formula: see text], where [Formula: see text] is a finite abelian group and [Formula: see text] is the univariate polynomial ring over a finite field [Formula: see text] of characteristic [Formula: see text]

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].

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