The unit group of finite group algebra of a generalized dihedral group

2014 ◽  
Vol 07 (02) ◽  
pp. 1450034 ◽  
Author(s):  
Neha Makhijani ◽  
R. K. Sharma ◽  
J. B. Srivastava
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Finite Field ◽  
Group Algebra ◽  
Dihedral Group ◽  
Unit Group ◽  
Generalized Dihedral Group

Let [Formula: see text] be a generalized dihedral group of order 2n and 𝔽q be a finite field having q elements. In this note, we establish the structure of the unit group of [Formula: see text] for any odd n ≥ 3. This extends a result due to Kaur and Khan [Units in 𝔽2D2p, J. Algebra Appl. 13(2) (2014) 9pp., doi: 10.1142/S0219498813500904] as well as a result due to the authors [Units in 𝔽2kD2n, Int. J. Group Theory 3(3) (2014) 25–34].

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].

The Structure of the Unit Group of the Group Algebra

2011 ◽  
Vol 54 (2) ◽  
pp. 237-243 ◽  
Author(s):  
Leo Creedon ◽  
Joe Gildea
Keyword(s):  
Finite Field ◽  
Group Ring ◽  
Group Algebra ◽  
Dihedral Group ◽  
Unit Group ◽  
Cyclic Groups ◽  
Split Extensions

AbstractLet RG denote the group ring of the group G over the ring R. Using an isomorphism between RG and a certain ring of n×n matrices in conjunction with other techniques, the structure of the unit group of the group algebra of the dihedral group of order 8 over any finite field of chracteristic 2 is determined in terms of split extensions of cyclic groups.

Structure of unit group of FpnD60

2020 ◽  
pp. 2150075
Author(s):  
Suchi Bhatt ◽  
Harish Chandra
Keyword(s):  
Finite Field ◽  
Group Algebra ◽  
Dihedral Group ◽  
Unit Group

Let [Formula: see text] be a finite field with characteristic [Formula: see text] having [Formula: see text] elements and [Formula: see text] be the dihedral group of order [Formula: see text]. In this paper, we have obtained the structure of unit groups of group algebra [Formula: see text], for any prime [Formula: see text].

Unit groups of finite group algebras of Abelian groups of order at most 16

2020 ◽  
pp. 2150030
Author(s):  
Meena Sahai ◽  
Sheere Farhat Ansari
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Finite Field ◽  
Group Algebra ◽  
Abelian Groups ◽  
Unit Group ◽  
Group Algebras

In this paper, we establish the structure of the unit group of the group algebra [Formula: see text] where [Formula: see text] is an abelian group of order at most 16 and [Formula: see text] is a finite field of characteristic [Formula: see text] with [Formula: see text] elements.

scholarly journals Group algebras whose p-elements form a subgroup

2016 ◽  
Vol 16 (09) ◽  
pp. 1750170
Author(s):  
M. Ramezan-Nassab
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Group Algebra ◽  
Polynomial Identity ◽  
Matrix Polynomial ◽  
Unit Group ◽  
Group Algebras ◽  
Locally Finite ◽  
P Elements

Let [Formula: see text] be a group, [Formula: see text] a field of characteristic [Formula: see text], and [Formula: see text] the unit group of the group algebra [Formula: see text]. In this paper, among other results, we show that if either (1) [Formula: see text] satisfies a non-matrix polynomial identity, or (2) [Formula: see text] is locally finite, [Formula: see text] is infinite and [Formula: see text] is an Engel-by-finite group, then the [Formula: see text]-elements of [Formula: see text] form a (normal) subgroup [Formula: see text] and [Formula: see text] is abelian (here, of course, [Formula: see text] if [Formula: see text]).

GROUP ALGEBRAS WITH UNIT GROUPS OF DERIVED LENGTH THREE

2010 ◽  
Vol 09 (02) ◽  
pp. 305-314 ◽  
Author(s):  
HARISH CHANDRA ◽  
MEENA SAHAI
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Sufficient Conditions ◽  
Unit Group ◽  
Necessary And Sufficient Conditions ◽  
Group Algebras ◽  
Characteristic P ◽  
Derived Length ◽  
A Finite Group ◽  
Necessary And Sufficient

Let K be a field of characteristic p ≠ 2,3 and let G be a finite group. Necessary and sufficient conditions for δ3(U(KG)) = 1, where U(KG) is the unit group of the group algebra KG, are obtained.

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]

scholarly journals On Clifford's theorem and ramification indices for symplectic modules over a finite field

1987 ◽  
Vol 30 (1) ◽  
pp. 153-167 ◽  
Author(s):  
Robert W. Van Der Waall
Keyword(s):  
Finite Group ◽  
Finite Field ◽  
Group Algebra ◽  
A Finite Group

Let K be a field, G a finite group. Let V be an (irreducible) KG-module, where KG is the group algebra consisting of all formal sums . The action of on α = ∑aθg on element ν ∈ V obeys the rule If H is a subgroup of G, then, restricting the action of G on V to H, V is also a KH-module. Notation: VH.

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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