Group algebras with almost minimal Lie nilpotency index

Let K be a field of positive characteristic p and KG the group algebra of a group G. It is known that if KG is Lie nilpotent, then its upper (or lower) Lie nilpotency index is at most |G′| + 1, where |G′| is the order of the commutator subgroup. The class of groups G for which these indices are maximal or almost maximal has already been determined. Here we determine G for which upper (or lower) Lie nilpotency index is the next highest possible.

Let [Formula: see text] be a group and let [Formula: see text] be a field of characteristic [Formula: see text]. Lie nilpotent group algebras of strong Lie nilpotency index at most 13 have been classified by many authors. In this paper, our aim is to classify the group algebras [Formula: see text] which are strongly Lie nilpotent of index 14.

Let [Formula: see text] be an arbitrary group and let [Formula: see text] be a field of characteristic [Formula: see text]. In this paper, we give some improvements of the upper bound of the lower Lie nilpotency index [Formula: see text] of the group algebra [Formula: see text]. We also give improved bounds for [Formula: see text], where [Formula: see text] is the number of independent generators of the finite abelian group [Formula: see text]. Furthermore, we give a description of the Lie nilpotent group algebra [Formula: see text] with [Formula: see text] or [Formula: see text]. We also show that for [Formula: see text] and [Formula: see text], [Formula: see text] if and only if [Formula: see text], where [Formula: see text] is the upper Lie nilpotency index of [Formula: see text].

In this paper, we classify the modular group algebra [Formula: see text] of a group [Formula: see text] over a field [Formula: see text] of characteristic [Formula: see text] having upper Lie nilpotency index [Formula: see text] for [Formula: see text] and [Formula: see text]. Group algebras of upper Lie nilpotency index [Formula: see text] for [Formula: see text], have already been characterized completely.

The Jacobson Radical of Group Algebras

In this paper, we classify the modular group algebra [Formula: see text] of a group [Formula: see text] over a field [Formula: see text] of characteristic [Formula: see text] having upper Lie nilpotency index [Formula: see text]. The group algebra [Formula: see text] with [Formula: see text] has already been described.

Let FG be the group algebra of a group G over a field F and let * denote the canonical involution of FG induced by the map g→g −1 ,gG. Let Un(FG)={uFG|uu * =1} be the group of unitary units of FG. In case char F=0, we classify the torsion groups G for which Un(FG) satisfies a group identity not vanishing on 2-elements. Along the way we actually prove that, in characteristic 0, the unitary group Un(FG) does not contain a free group of rank 2 if FG − , the Lie algebra of skew elements of FG, is Lie nilpotent. Motivated by this connection we characterize most groups G for which FG − is Lie nilpotent and char F≠2.

Lie properties of symmetric elements in group rings

Let KG be the group algebra of a group G over a field K of characteristic p > 0. It is proved that the following statements are equivalent: KG is Lie nilpotent of class ≤ p, KG is strongly Lie nilpotent of class ≤ p and G′ is a central subgroup of order p. Also, if G is nilpotent and G′ is of order pn then KG is strongly Lie nilpotent of class ≤ pn and both U(KG)/ζ(U(KG)) and U(KG)′ are of exponent pn. Here U(KG) is the group of units of KG. As an application it is shown that for all n ≤ p+ 1, γn(L(KG)) = 0 if and only if γn(KG) = 0.