Modular group algebras with almost maximal Lie nilpotency indices. I

Modular group algebras with maximal Lie nilpotency indices

In the present paper we give the full description of the Lie nilpotent group algebras which have maximal Lie nilpotency indices.

模群代数的下与上李幂零指数

Lower and Upper Lie Nilpotency Indices of Modular Group Algebras

该研究解决了模群代数下与上李幂零指数问题的剩余情形，在特征2和3下证明两指数相等，完成有限域上素数幂阶有限群的相关问题，引入终端限制PBW构造完成论证。

On the Lie nilpotency index of modular group algebras

Let $KG$ be the modular group algebra of an arbitrary group $G$ over a field $K$ of characteristic $p>0$. It is seen that if $KG$ is Lie nilpotent, then its lower as well as upper Lie nilpotency index is at least $p+1$. The classification of group algebras $KG$ with upper Lie nilpotency index $t^{L}(KG)$ upto $9p-7$ have already been determined. In this paper, we classify the modular group algebra $KG$ for 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 $KG$ of a group $G$ over a field $K$ of characteristic $p>0$ having upper Lie nilpotency index $t^{L}(KG)= \vert G^{\prime}\vert - k(p-1) + 1$ for $k=14$ and $15$. Group algebras of upper Lie nilpotency index $\vert G^{\prime}\vert - k(p-1) + 1$ for $k\leq 13$, have already been characterized completely.

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 have 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, II

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. Previously we determined the groups G for which the upper/lower nilpotency index is maximal or the upper nilpotency index is `almost maximal' (that is, of the next highest possible value, namely |G'|-p +2). Here we determine the groups for which the lower nilpotency index is `almost maximal'.