Antisymmetric Elements in Group Rings Ii

Cristo, Osnel Broche; Jespers, Eric; Milies, César Polcino; Marín, Manuel Ruiz

doi:10.1142/s0219498809003254

Cited by 13 publications

(4 citation statements)

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“…For example, the conditions under which the skew elements commute were determined in Broche et al [4]. Subsequently, Giambruno et al [7] determined the torsion groups without 2-elements such that (FG) − is Lie nilpotent, and Catino et al handled the bounded Lie Engel property in [5].…”

Section: Introductionmentioning

confidence: 99%

Lie Metabelian Skew Elements in Group Rings

Lee

Spinelli

2013

Glasgow Math. J.

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Let F be a field of characteristic p = 2 and G a group without 2-elements having an involution * . Extend the involution linearly to the group ring FG, and let (FG) − denote the set of skew elements with respect to * . In this paper, we show that if G is finite and (FG) − is Lie metabelian, then G is nilpotent. Based on this result, we deduce that if G is torsion, p > 7 and (FG) − is Lie metabelian, then G must be abelian. Exceptions are constructed for smaller values of p.2010 Mathematics Subject Classification. 16S34, 16R50. Introduction.Let FG be the group ring of a group G over a field F of characteristic p = 2. If G has an involution * , then we extend it linearly to obtain an involution of FG, also denoted by * . We write (FG) − for the set of all skew elements of FG, that is, those elements α satisfying α * = −α. It is easy to see that (FG) − is the Lie sub-algebra of FG consisting of all linear combinations of terms of the form g − g * , with g ∈ G. A general problem in group rings is to decide the extent to which the structure of (FG) − determines the structure of FG. If g * = g −1 for all g ∈ G, then the induced involution is called the classical involution. During the last two decades, a considerable amount of attention has been devoted to determining if Lie properties satisfied by (FG) − are also satisfied by FG. Giambruno and Sehgal in [8] showed that if G has no 2-elements and (FG) − is Lie nilpotent, so is FG. Lee [9] proved the analogous result for the bounded Lie Engel property. Lie solvability was considered by Lee et al.[11] with suitable restrictions upon G. More recently, a great deal of work has appeared considering involutions other than the classical one. For example, the conditions under which the skew elements commute were determined in Broche et al. [4]. Subsequently, Giambruno et al. [7] determined the torsion groups without 2-elements such that (FG) − is Lie nilpotent, and Catino et al. handled the bounded Lie Engel property in [5]. We note that the situation with the skew elements is more involved than the corresponding problem for the symmetric elements. Indeed, the absence of 2-elements was not sufficient to force FG to have the same property in these papers; there were exceptional cases.This type of work has inspired similar investigations in other specific classes of algebras with involution. For instance, Siciliano [17] has characterized restricted Lie

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Section: Introductionmentioning

confidence: 99%

Lie Metabelian Skew Elements in Group Rings

Lee

Spinelli

2013

Glasgow Math. J.

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“…In [2], for instance, Broche Cristo, Jespers, Polcino Milies and Ruiz Marín determined when the skew elements of F G commute, for any involution induced from an involution on G. Subsequently, in [6], Giambruno, Polcino Milies and Sehgal determined when (F G) − is Lie nilpotent, if G is a torsion group having no elements of order 2. Catino, Lee and Spinelli [3] proved the corresponding result for the bounded Lie Engel property.…”

Section: Introductionmentioning

confidence: 99%

Group rings whose skew elements are bounded Lie Engel

Lee

Spinelli

2015

Journal of Pure and Applied Algebra

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“…For example, in [7], Jespers and Ruiz Marín determined when (FG) + is commutative; subsequently, in [1], Broche Cristo et al answered the same question for (FG) − . In Giambruno et al [2] and Lee et al [10], the conditions under which (FG) + is Lie nilpotent or bounded Lie Engel were determined for an arbitrary involution on G. In particular, if G has no elements of order 2, then the same result holds as for the classical involution.…”

Section: Introductionmentioning

confidence: 99%