On symmetric units in group algebras

Abstract

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.