Nonunits of group algebras over the fours group

2019 ◽  
Vol 18 (12) ◽  
pp. 1950228 ◽  
Author(s):  
A. Abdollahi ◽  
S. M. Zanjanian
Keyword(s):  
Group Algebra ◽  
Nonzero Element ◽  
Group Algebras ◽  
Supersoluble Group ◽  
Wide Range ◽  
Torsion Free

The conjecture on units of group algebras of a torsion-free supersoluble group is saying that every unit is trivial, i.e. a product of a nonzero element of the field and an element of the group. This conjecture is still open and even in the slightly simple case of the fours group [Formula: see text], it is not yet known. The main result of this paper is to show that a wide range of elements of group algebra of [Formula: see text] are nonunit.

scholarly journals Group algebras of some generalised soluble groups

1972 ◽  
Vol 18 (1) ◽  
pp. 1-5 ◽  
Author(s):  
R. P. Knott
Keyword(s):  
Free Group ◽  
Group Algebra ◽  
Soluble Group ◽  
Group Algebras ◽  
Soluble Groups ◽  
Torsion Free Group ◽  
Open Question ◽  
Torsion Free

In (8) Stonehewer referred to the following open question due to Amitsur: If G is a torsion-free group and F any field, is the group algebra, FG, of G over F semi-simple? Stonehewer showed the answer was in the affirmative if G is a soluble group. In this paper we show the answer is again in the affirmative if G belongs to a class of generalised soluble groups

scholarly journals Free Group Algebras in Division Rings with Valuation II

2019 ◽  
Vol 72 (6) ◽  
pp. 1463-1504
Author(s):  
Javier Sánchez
Keyword(s):  
Free Group ◽  
Group Algebra ◽  
Neumann Series ◽  
Group Algebras ◽  
Series Ring ◽  
Free Nilpotent Group ◽  
Enveloping Algebra ◽  
Division Rings ◽  
Ore Domain ◽  
Torsion Free

AbstractWe apply the filtered and graded methods developed in earlier works to find (noncommutative) free group algebras in division rings.If $L$ is a Lie algebra, we denote by $U(L)$ its universal enveloping algebra. P. M. Cohn constructed a division ring $\mathfrak{D}_{L}$ that contains $U(L)$. We denote by $\mathfrak{D}(L)$ the division subring of $\mathfrak{D}_{L}$ generated by $U(L)$.Let $k$ be a field of characteristic zero, and let $L$ be a nonabelian Lie $k$-algebra. If either $L$ is residually nilpotent or $U(L)$ is an Ore domain, we show that $\mathfrak{D}(L)$ contains (noncommutative) free group algebras. In those same cases, if $L$ is equipped with an involution, we are able to prove that the free group algebra in $\mathfrak{D}(L)$ can be chosen generated by symmetric elements in most cases.Let $G$ be a nonabelian residually torsion-free nilpotent group, and let $k(G)$ be the division subring of the Malcev–Neumann series ring generated by the group algebra $k[G]$. If $G$ is equipped with an involution, we show that $k(G)$ contains a (noncommutative) free group algebra generated by symmetric elements.

Zero divisors and idempotents in group rings

1977 ◽  
Vol 81 (3) ◽  
pp. 365-368 ◽  
Author(s):  
P. A. Linnell
Keyword(s):  
Group Algebra ◽  
Zero Divisor ◽  
Affirmative Answer ◽  
Supersoluble Group ◽  
Group Rings ◽  
Prime Characteristic ◽  
Locally Finite ◽  
Zero Divisors ◽  
Torsion Free

1. Introduction. Let kG denote the group algebra of a group G over a field k. In this paper we are first concerned with the zero divisor conjecture: that if G is torsion free, then kG is a domain. Recently, K. A. Brown made a remarkable breakthrough when he settled the conjecture for char k = 0 and G abelian by finite (2). In a beautiful paper written shortly afterwards, D. Farkas and R. Snider extended this result to give an affirmative answer to the conjecture for char k = 0 and G polycyclic by finite. Their methods, however, were less successful when k had prime characteristic. We use the techniques of (4) and (7) to prove the following:Theorem A. If G is a torsion free abelian by locally finite by supersoluble group and k is any field, then kG is a domain.

scholarly journals Cardinality of product sets in torsion-free groups and applications in group algebras

2019 ◽  
Vol 19 (04) ◽  
pp. 2050079
Author(s):  
Alireza Abdollahi ◽  
Fatemeh Jafari
Keyword(s):  
Group Theory ◽  
Free Group ◽  
Group Algebra ◽  
Finite Subset ◽  
Identity Element ◽  
Klein Bottle ◽  
Group Algebras ◽  
Torsion Free Group ◽  
Product Sets ◽  
Torsion Free

Let [Formula: see text] be a unique product group, i.e. for any two finite subsets [Formula: see text] of [Formula: see text], there exists [Formula: see text] which can be uniquely expressed as a product of an element of [Formula: see text] and an element of [Formula: see text]. We prove that if [Formula: see text] is a finite subset of [Formula: see text] containing the identity element such that [Formula: see text] is not abelian, then, for all subsets [Formula: see text] of [Formula: see text] with [Formula: see text], [Formula: see text]. Also, we prove that if [Formula: see text] is a finite subset containing the identity element of a torsion-free group [Formula: see text] such that [Formula: see text] and [Formula: see text] is not abelian, then for all subsets [Formula: see text] of [Formula: see text] with [Formula: see text], [Formula: see text]. Moreover, if [Formula: see text] is not isomorphic to the Klein bottle group, i.e. the group with the presentation [Formula: see text], then for all subsets [Formula: see text] of [Formula: see text] with [Formula: see text], [Formula: see text]. The support of an element [Formula: see text] in a group algebra [Formula: see text] ([Formula: see text] is any field), denoted by [Formula: see text], is the set [Formula: see text]. By the latter result, we prove that if [Formula: see text] for some nonzero [Formula: see text] such that [Formula: see text], then [Formula: see text]. Also, we prove that if [Formula: see text] for some [Formula: see text] such that [Formula: see text], then [Formula: see text]. These results improve a part of results in Schweitzer [J. Group Theory 16(5) (2013) 667–693] and Dykema et al. [Exp. Math. 24 (2015) 326–338] to arbitrary fields, respectively.

Complex group algebras of the direct product of non-isomorphic Suzuki groups

2019 ◽  
Vol 19 (02) ◽  
pp. 2050036
Author(s):  
Morteza Baniasad Azad ◽  
Behrooz Khosravi
Keyword(s):  
Direct Product ◽  
Group Algebra ◽  
Irreducible Character ◽  
Prime Numbers ◽  
Product Formula ◽  
Character Degrees ◽  
Group Algebras ◽  
Complex Group ◽  
Complex Group Algebra ◽  
Suzuki Groups

In this paper, we prove that the direct product [Formula: see text], where [Formula: see text] are distinct numbers, is uniquely determined by its complex group algebra. Particularly, we show that the direct product [Formula: see text], where [Formula: see text]’s are distinct odd prime numbers, is uniquely determined by its order and three irreducible character degrees.

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].

COMMUTATOR IDENTITIES ON SYMMETRIC ELEMENTS OF GROUP ALGEBRAS

2013 ◽  
Vol 12 (08) ◽  
pp. 1350044
Author(s):  
TIBOR JUHÁSZ ◽  
ENIKŐ TÓTH
Keyword(s):  
Direct Product ◽  
Group Algebra ◽  
Group Algebras ◽  
Augmentation Ideal ◽  
Characteristic P ◽  
Derived Length ◽  
Commutator Identity ◽  
Nilpotency Index ◽  
Powerful Group

Let K be a field of odd characteristic p, and let G be the direct product of a finite p-group P ≠ 1 and a Hamiltonian 2-group. We show that the set of symmetric elements (KG)* of the group algebra KG with respect to the involution of KG which inverts all elements of G, satisfies all Lie commutator identities of degree t(P) or more, where t(P) denotes the nilpotency index of the augmentation ideal of the group algebra KP. In addition, if P is powerful, then (KG)* satisfies no Lie commutator identity of degree less than t(P). Applying this result we get that (KG)* is Lie nilpotent and Lie solvable, and its Lie nilpotency index and Lie derived length are not greater than t(P) and ⌈ log 2 t(P)⌉, respectively, and these bounds are attained whenever P is a powerful group. The corresponding result on the set of symmetric units of KG is also obtained.

Group Algebras with Minimal Strong Lie Derived Length

2008 ◽  
Vol 51 (2) ◽  
pp. 291-297 ◽  
Author(s):  
Ernesto Spinelli
Keyword(s):  
Solvable Group ◽  
Group Algebra ◽  
Positive Characteristic ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Algebras ◽  
Derived Length ◽  
Necessary And Sufficient

AbstractLet KG be a non-commutative strongly Lie solvable group algebra of a group G over a field K of positive characteristic p. In this note we state necessary and sufficient conditions so that the strong Lie derived length of KG assumes its minimal value, namely [log2(p + 1)].

WARFIELD INVARIANTS IN COMMUTATIVE GROUP ALGEBRAS

2008 ◽  
Vol 07 (03) ◽  
pp. 337-346 ◽  
Author(s):  
PETER V. DANCHEV
Keyword(s):  
Abelian Group ◽  
Prime Number ◽  
Group Algebra ◽  
Ordinal Number ◽  
Commutative Group ◽  
Group Algebras

Let F be a field and G an Abelian group. For every prime number q and every ordinal number α we compute only in terms of F and G the Warfield q-invariants Wα, q(VF[G]) of the group VF[G] of all normed units in the group algebra F[G] under some minimal restrictions on F and G. This expands own recent results from (Extracta Mathematicae, 2005) and (Collectanea Mathematicae, 2008).

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