Units of Group Algebras of the Fours Group
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Units of Group Algebras of the Fours Group Alireza Abdollahi1,2 · Soraya Mahdi Zanjanian1 Received: 24 February 2019 / Revised: 31 October 2019 / Accepted: 5 November 2019 © Iranian Mathematical Society 2019
Abstract We obtain some necessary conditions on coefficients of possible units of even (resp., odd) L-length in the group algebra K , where is the (Passman) fours group. In particular, the relations between the coefficients of possible units of L-length greater or equal to 4, in this group algebra, are also investigated. Keywords Torsion-free group · Group algebras · Fours group · Unit conjecture Mathematics Subject Classification 16S34 · 16U60 · 20C05 · 20C07
1 Introduction and Results For a torsion-free group G and a field K a trivial unit in the group algebra K G is of the form k · g for some k ∈ K \{0} and g ∈ G. The famous unit conjecture states that [9,11,13,14]: Conjecture 1.1 For a torsion-free group G and a field K all units of the group algebra K G are trivial. There is another famous conjecture on the zero divisors of the group algebra K G as follows. Conjecture 1.2 Let K be a field and G a torsion-free group. The group algebra K G has no zero divisor, i.e., if αβ = 0 for two elements α and β in K G, then α = 0 or β = 0.
Communicated by Mohammad Reza Darafsheh.
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Alireza Abdollahi [email protected] Soraya Mahdi Zanjanian [email protected]; [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
In the USSR the Zero Divisor Conjecture was referred to as “the Malcev’s problem” during a certain period of time and it is well-known as Kaplansky Zero Divisor Conjecture too. A. A. Bovdi was the first who suggested to study Conjecture 1.1 so-called Unit Conjecture (see [2,3]). It is known that if there is a zero divisor in K G then there is a non-trivial unit in K G [11, p. 584, Lemma 1.2]. In 1973, Formanek [6] using a theorem of Lewin [8] proved that torsion-free supersoluble groups satisfy Conjecture 1.2. Torsion-free virtually soluble groups satisfy the zero divisor conjecture. The latter was shown by Kropholler, Linnell and Moody in 1988 [7]. The unit conjecture is valid for unique-product groups [11,12,15] (a group G 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 ab, where a ∈ X and b ∈ Y ). The class of unique-product groups contains all orderable and locally indicable groups, so Conjecture 1.1 is valid for them and therefore all torsionfree nilpotent groups satisfy Conjecture 1.1. Pappas in [10] relates the unique-product property to support set of ∗-symmetric units in group algebras. Conjecture 1.1 is still open for supersoluble groups G. By induction on the Hirsch number of G using a similar argument as in [11, p. 615, Theorem 3.9] only the case in
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