Quotients of Passman Fours Group and Non-units of Their Group Algebras
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Quotients of Passman Fours Group and Non-units of Their Group Algebras Alireza Abdollahi1,2 · Soraya Mahdi Zanjanian1 Received: 10 June 2020 / Accepted: 3 October 2020 © Iranian Mathematical Society 2020
Abstract The famous unit conjecture for group algebras states that every unit is trivial. The validity of this conjecture is not known for the sightly simple example of fours group = x, y | (x 2 ) y = x −2 , (y 2 )x = y −2 which it is “the simplest” example of a torsion-free non unique-product supersoluble group. In this article for n ∈ N, we set n n n Hn = x 2 , y 2 , (x y)2 ≤ and we consider G n = /Hn . We will show that there is a large subset Nn of C[G n ] which its elements are non-unit, so all elements of the set N = n∈N ϕn−1 (Nn ) are non-unit in C[], where ϕn : C[] → C[G n ] is the induced group ring homomorphism by the quotient map ϕn : → G n . Keywords Torsion-free group · Group algebras · Fours group · Unit conjecture Mathematics Subject Classification 16S34 · 16U60 · 20C05 · 20C07
1 Introduction and Results Let G be a torsion-free group and K be a field. The unit conjecture and the zero divisor conjecture are two famous conjectures on group algebra K [G]. The following is known as The Unit Conjecture [3,11,12].
Communicated by Mohammad Shahryari.
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Soraya Mahdi Zanjanian [email protected]; [email protected] Alireza Abdollahi [email protected]
1
Department of Mathematics, University of Isfahan, Isfahan 81746-73441, Iran
2
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran
123
Bulletin of the Iranian Mathematical Society
Conjecture 1.1 Let G be a torsion-free group and K be a field. Then, all units of the group algebra K [G] are trivial. Indeed, the units are of the form α · g for some α ∈ K \{0} and g ∈ G. The Zero Divisor Conjecture states that: Conjecture 1.2 For a torsion-free group G and a field K , the group algebra K [G] has no proper zero divisor, i.e., for two elements α and β in K [G], if αβ = 0, then α = 0 or β = 0. The first who suggested to study unit conjecture was A. A. Bovdi (see [4,5]). It is known that the validity of Conjecture 1.1 implies the validity of Conjecture 1.2 [11, p. 584, Lemma 1.2]. Formanek [7] using a theorem of Lewin [9] improved Bovdi’s result concerning Conjecture 1.2 to torsion-free supersoluble groups. In 1988, Kropholler et al. have shown that torsion-free soluble-by-finite groups satisfy the zero divisor conjecture [8]. Conjecture 1.1 is valid for unique-product groups [11,12,14]. (A group is called unique product if, for any two finite non-empty subsets X and Y of the group, there exists an element g in the group having a unique representation of the form x y, where (x, y) ∈ X × Y ). Because class of unique-product groups contains all right or left orderable and locally indicable groups, so Conjecture 1.1 is valid for them, and therefore, it is true for all torsion-free nilpotent groups. Pappas in [10] relates the unique-product property to support set of ∗-sym
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