Lie Nilpotency Index Research Articles

A note on modular group algebras with upper Lie nilpotency indices

Let KG be the modular group algebra of anarbitrary group G over a field K of characteristic p>0. In thispaper we give some improvements of upper Lie nilpotency indext L(KG) of the group algebra KG. It can be seen that if KG is Lie nilpotent, then its lower as well as upper Lie nilpotency index is atleast p+1. In this way the classification of group algebras KG with next upper Lie nilpotency indext L(KG) up to 9p−7 have alreadybeen classified. Furthermore, we give a complete classification ofmodular group algebraKGfor which the upper Lie nilpotency index is 10p−8.

Lie nilpotency index of a modular group algebra

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.

On the lower Lie nilpotency index of a group algebra

In this article, we show that if KG is a Lie nilpotent group algebra of a group G over a field K of characteristic p > 0, then tL (KG) = k if and only if tL (KG) = k, for k ∈ {5p − 3, 6p − 4}, where tL (KG) and tL (KG) are the lower and the upper Lie nilpotency indices of KG, respectively.

Modular Group Algebras with Small Upper Lie Nilpotency Index

Let KG be the modular group algebra of a group G over a field K of characteristic p > 0. The classification of group algebras KG with upper Lie nilpotency index tL(KG) greater than or equal to |G′| – 13p + 14 have already been done. In this paper, our aim is to classify the group algebras KG for which tL(KG) = |G′| – 14p + 15.

A note on the upper Lie nilpotency index of a group algebra

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.

A note on Lie nilpotent group algebras

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].

Group algebras of Lie nilpotency index 14

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.

Lie nilpotency indices of modular group algebras II

Let [Formula: see text] be the modular group algebra of a group [Formula: see text] over a field [Formula: see text] of characteristic [Formula: see text]. The classification of the group algebras [Formula: see text] with upper Lie nilpotency index [Formula: see text] greater than or equal to [Formula: see text] has already been done. In this paper, we determine the group algebras [Formula: see text] such that [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text].

Modular group algebras of Lie nilpotency index 8p − 6

For a group algebra [Formula: see text] of a group [Formula: see text] over a field [Formula: see text] of characteristic [Formula: see text], it is well known that [Formula: see text] is the minimal upper as well as the minimal lower Lie nilpotency index. Group algebras of upper Lie nilpotency index upto [Formula: see text] have already been characterized completely. In this paper, we classify the modular group algebra [Formula: see text] having upper Lie nilpotency index [Formula: see text] which is the possible next higher Lie nilpotency index.

Group algebras of Lie nilpotency index 12 and 13

ABSTRACTLet G be a group and let K be a field of characteristic p>0. Lie nilpotent group algebras of strong Lie nilpotency index up to 11 have already been classified. In this paper, our aim is to classify the group algebras KG which are strongly Lie nilpotent of index 12 or 13.

On Lie nilpotent modular group algebras

ABSTRACTLet K be a field of characteristic p>0 and let KG be the group algebra of an arbitrary group G over K. It is known that if KG is Lie nilpotent, then its lower as well as upper Lie nilpotency index is at least p+1. The group algebras KG for which these indices are p+1 or 2p or 3p−1 or 4p−2 have already been determined. In this paper, we classify the group algebras KG for which the upper Lie nilpotency index is 5p−3, 6p−4 or 7p−5.

Modular group algebras with small Lie nilpotency indices

Let [Formula: see text] be a group and let [Formula: see text] be a field of characteristic [Formula: see text]. In this paper, we have classified the group algebras [Formula: see text] which are strongly Lie nilpotent of index [Formula: see text], [Formula: see text] or [Formula: see text].

On the Lie nilpotency indices of modular group algebras

Let KG be the group algebra of a group G over a field K of characteristic \(p>0\). The classification of group algebras KG for which Lie nilpotency indices are maximal or almost maximal has already been determined. After that Bovdi and Srivastava classified Lie nilpotent group algebras with \(t^{L}(KG)=|G^{\prime }| - 2p +3,|G^{\prime }| - 3p +4\) and \( |G^{\prime }| - 4p +5.\) In this paper, our aim is to classify the group algebras KG for which \(t^{L}(KG)=|G^{\prime }| - 5p +6,|G^{\prime }| - 6p +7\) and \( |G^{\prime }| - 7p +8.\)

Group algebras with almost minimal Lie nilpotency index

Group algebras with almost minimal Lie nilpotency index

COMMUTATOR IDENTITIES ON SYMMETRIC ELEMENTS OF GROUP ALGEBRAS

Let K be a field of odd characteristic p, and let G be the direct product of a finite p-group P ≠ 1 and a Hamiltonian 2-group. We show that the set of symmetric elements (KG)* of the group algebra KG with respect to the involution of KG which inverts all elements of G, satisfies all Lie commutator identities of degree t(P) or more, where t(P) denotes the nilpotency index of the augmentation ideal of the group algebra KP. In addition, if P is powerful, then (KG)* satisfies no Lie commutator identity of degree less than t(P). Applying this result we get that (KG)* is Lie nilpotent and Lie solvable, and its Lie nilpotency index and Lie derived length are not greater than t(P) and ⌈ log 2 t(P)⌉, respectively, and these bounds are attained whenever P is a powerful group. The corresponding result on the set of symmetric units of KG is also obtained.

Lie nilpotency indices of symmetric elements under oriented involutions in group algebras

Let G be a group and let F be a eld of characteristic dierent from 2. Denote by (FG)+ the set of symmetric elements and by U+(FG) the set of symmetric units, under an oriented classical involution of the group algebra FG. We give some lower and upper bounds on the Lie nilpotency index of (FG)+ and the nilpotency class of U+(FG).

Nilpotency Indices of Symmetric Elements of Group Algebras

Let G be a group and let F be a field of characteristic different from 2. Denote by (FG)+ the set of symmetric elements and by U +(FG) the set of symmetric units of the group algebra FG, with respect to the canonical involution on FG sending each group element to its inverse. In this article, we give some lower and upper bounds on the Lie nilpotency index of (FG)+ and the nilpotency class of U +(FG); furthermore, we classify the groups for which these bounds are achieved.

Lie Nilpotency Indices of Modular Group Algebras

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.

Modular Group Algebras with Almost Maximal Lie Nilpotency Indices

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 (and lower) Lie nilpotency index is at most \(|G^{\, \prime}|+1\), where \(|G^{\, \prime}|\) is the order of the commutator subgroup. The authors previously determined those groups \(G\) for which this index is maximal and here they determine the groups \(G\) for which it is `almost maximal', that is, it takes the next highest possible value, namely \(|G^{\, \prime}|-p+2\).

Lie Nilpotency Indices of Restricted Universal Enveloping Algebras

ABSTRACT For a restricted Lie algebra L over a field of characteristic p > 0 we study the Lie nilpotency index t L (u(L)) of its restricted universal enveloping algebra u(L). In particular, we determine an upper and a lower bound for t L (u(L)). Finally, under the assumption that L is p-nilpotent and finite-dimensional, we establish when the Lie nilpotency index of u(L) is maximal. Communicated by I. Shestakov.