1,721,017 research outputs found
Group algebras with minimal strong Lie derived length
No full textLet 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 [log(2) (P + 1)]
Group algebras with minimal Lie derived length
No full textLet KG be a non-commutative Lie solvable group algebra of a group G over a field K of positive characteristic p. In [A. Shalev, The derived length of Lie soluble group rings I, J. Pure Appl. Algebra 78 (1992) 291-300] Shalev proved that dl(L) (KG) >= [log(2) (P + 1)] and posed the question of characterizing group algebras for which this lower bound is achieved. In this note the solution to this question is given. (C) 2008 Elsevier Inc. All rights reserved
Lie solvable group algebras and solvable unit group
No full textThis note is a short survey of recent results and open problems in the framework of Lie solvable group algebras
Lie Dimension Subgroups and Central Series Related to Group Algebras
No full textLet KG be the group algebra of a group G over a field K of positive characteristic p, and let D((n)) (G) and D([n]) (G) denote the n-th upper Lie dimension subgroup and the n-th lower one, respectively. In [1] and [12], the equality D((n)) (G) = D([n]) (G) is verified when p >= 5. Motivated by [16, Problem 55], in the present paper we establish it for particular classes of groups when p <= 3. Finally, we introduce and study a new central series of G linked with the Lie nilpotency class of KG
Group algebras with almost maximal lie nilpotency index
No full textLet KG be a non-commutative Lie nilpotent group algebra of a group G over a field K. It is known that the Lie nilpotency index of KG is at most {pipe}G′{pipe} + 1, where {pipe}G′{pipe} is the order of the commutator subgroup of G. In [4] the groups G for which this index is maximal were determined. Here we list the G's for which it assumes the next highest possible value. © 2005 Springer
Group rings with metabelian unit groups
Full text linkLet F be a field of odd characteristic and G a group. In 1991 Shalev established necessary and sufficient conditions so that the unit group of the group ring FG is metabelian when G is finite. Here, in the modular case, we do the same without restrictions on G. In particular, new cases emerge when G contains elements of infinite orde
Group rings whose skew elements are bounded Lie Engel
Full text linkLet FG be the group ring of a group G over a field F of characteristic different from 2, and let FG have an involution induced from one on G. Assuming that G has no elements of order 2 and no dihedral group involved, we determine the conditions under which the set of skew elements of FG is bounded Lie Engel. Furthermore, we make the determination with no restrictions upon G when the involution on FG is classical
Lie metabelian skew elements in group rings
No full textLet 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
Minimal varieties of associative PI (super)-algebras with respect to their (graded) exponent
No full textGiambruno and Zaicev (Trans Am Math Soc 355: 5091-5117, 2003) characterized varieties of associative PI-algebras over a field of characteristic zero which are minimal of fixed exponent. The aim of the present survey paper is to discuss this result and the recent developments on the corresponding problem for PI-algebras endowed with a Z_2-grading. In particular, we provide an example of minimal superalgebra not generating a minimal supervariety of fixed superexponent, thereby partially answering the question of which actually are the generators of minimal supervarieties of finite basic rank
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