Narrow Results

Let $G$ be an arbitrary group and let $K$ be a field of characteristic $p>0$. In this paper, we give some improvements of the upper bound of the lower Lie nilpotency index $t_L(KG)$ of the group algebra $KG$. We also give improved bounds for $m_j$, where $m_j$ is the number of independent generators of the finite abelian group ${\gamma }_j(G)/{\gamma }_{j+1}(G)$. Furthermore, we give a description of the Lie nilpotent group algebra $KG$ with $t_L(KG)=7$ or $8$. We also show that for $k=7$ and $8$, $t_L(KG)=k$ if and only if $t^L(KG)=k$, where $t^L(KG)$ is the upper Lie nilpotency index of $KG$.

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.

For a group algebra $KG$ of a group $G$ over a field $K$ of characteristic $p>0$, it is well known that $p+1$ is the minimal upper as well as the minimal lower Lie nilpotency index. Group algebras of upper Lie nilpotency index upto $7p-5$ have already been characterized completely. In this paper, we classify the modular group algebra $KG$ having upper Lie nilpotency index $8p-6$ which is the possible next higher Lie nilpotency index.

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 $t^L(KG)=9p-7$. The group algebra $KG$ with $t^L(KG)<9p-7$ has already been described.

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 $t^L(KG)=\left|G^{\prime }\right|-k(p-1)+1$ for $k=14$ and $15$. Group algebras of upper Lie nilpotency index $\left|G^{\prime }\right|-k(p-1)+1$ for $k\le 13$, have already been characterized completely.