Let $KG$ be the group algebra of a group $G$ over a field $K$ of positive characteristic $p$ and let $\mathfrak D_{(n)}(G)$ and $\mathfrak D_{[n]}(G)$ denote the $n$-th upper Lie dimension subgroup and the $n$-th lower one, respectively. Bhandari and Passi [\emph{Residually Lie nilpotent group rings}, Arch. Math. {\bf{58}} (1992), 1-6] and, independently, Riley [\emph{Restricted Lie dimension subgroups}, Comm. Algebra {\bf{19}} (1991), 1493-1499] verified the equality $\mathfrak D_{(n)}(G)=\mathfrak D_{[n]}(G)$ when $p\geq 5$. Motivated by Problem n. 55 of [Sehgal, S.K.:\emph{Units in integral group rings}, Pitman monographs and surveys in pure and applied mathematics, New York, 1993], in the present note we establish it for particular classes of groups, when $p\leq 3$. Finally, we introduce and study a new central series of $G$ linked with the Lie nilpotency class of $KG$.
Lie dimension subgroups and central series related to group algebras
SPINELLI, Ernesto
2009-01-01
Abstract
Let $KG$ be the group algebra of a group $G$ over a field $K$ of positive characteristic $p$ and let $\mathfrak D_{(n)}(G)$ and $\mathfrak D_{[n]}(G)$ denote the $n$-th upper Lie dimension subgroup and the $n$-th lower one, respectively. Bhandari and Passi [\emph{Residually Lie nilpotent group rings}, Arch. Math. {\bf{58}} (1992), 1-6] and, independently, Riley [\emph{Restricted Lie dimension subgroups}, Comm. Algebra {\bf{19}} (1991), 1493-1499] verified the equality $\mathfrak D_{(n)}(G)=\mathfrak D_{[n]}(G)$ when $p\geq 5$. Motivated by Problem n. 55 of [Sehgal, S.K.:\emph{Units in integral group rings}, Pitman monographs and surveys in pure and applied mathematics, New York, 1993], in the present note we establish it for particular classes of groups, when $p\leq 3$. Finally, we introduce and study a new central series of $G$ linked with the Lie nilpotency class of $KG$.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
