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'.Comment: 7 page

On symmetric units in group algebras

Let $U(KG)$ be the group of units of the group ring $KG$ of the group $G$ over a commutative ring $K$. The anti-automorphism g\mapsto g\m1 of $G$ can be extended linearly to an anti-automorphism $a\mapsto a^*$ of $KG$. Let $S_*(KG)=\{x\in U(KG) \mid x^*=x\}$ be the set of all symmetric units of $U(KG)$. We consider the following question: for which groups $G$ and commutative rings $K$ it is true that $S_*(KG)$ is a subgroup in $U(KG)$. We answer this question when either a) $G$ is torsion and $K$ is a commutative $G$-favourable integral domain of characteristic $p\geq 0$ or b) $G$ is non-torsion nilpotent group and $KG$ is semiprime.Comment: 11 pages, AMS-TeX, to appear in Comm. in Algebr

Symmetric units in modular group algebras

Let p be a prime, G a locally finite p-group, K a commutative ring of characteristic p. The anti-automorphism g bar arrow pointing right g(-1) of G extends linearly to an anti-automorphism a bar arrow pointing right a* of KG. An element a of KG is called symmetric if a* = a. In this paper we answer the question: for which G and K do the symmetric units of KG form a multiplicative group

On elements in algebras having finite number of conjugates

Let $R$ be a ring with unity and $U(R)$ its group of units. Let $\Delta U=\{a\in U(R)\mid [U(R):C_{U(R)}(a)]<\infty\}$ be the $FC$-radical of $U(R)$ and let $\nabla(R)=\{a\in R\mid [U(R):C_{U(R)}(a)]<\infty\}$ be the $FC$-subring of $R$. An infinite subgroup $H$ of $U(R)$ is said to be an $\omega$-subgroup if the left annihilator of each nonzero Lie commmutator $[x,y]$ in $R$ contains only finite number of elements of the form $1-h$, where $x,y \in R$ and $h\in H$. In the case when $R$ is an algebra over a field $F$, and $U(R)$ contains an $\omega$-subgroup, we describe its $FC$-subalgebra and the $FC$-radical. This paper is an extension of [1].Comment: 8 pages, AMS-Te

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