Some identities on twisted group algebras

Abstract

Let G be a group and F be a field of characteristic $p\ge 0$. In this paper we provide some necessary and sufficient conditions for a twisted group algebra $F^{\tau }G$ to satisfies a non-matrix identity. This allows us to show that a crossed product $F_{\sigma }^{\tau }G$ is Lie nilpotent if and only if σ is trivial, G is nilpotent and p-abelian, G has a unique normal Sylow p-subgroup P and $G=P\times Q$ for some central $p\prime$-subgroup Q and $F^{\tau }G$ is stably untwisted. Also, ${\mathbb{Z}}_{\sigma }^{\tau }G$ is Lie nilpotent if and only if ${\mathbb{Z}}_{\sigma }^{\tau }G$ is commutative. Some generalized group identities on the unit group of $\mathrm{(}\text{HTML}\mathrm{ }\text{translation}\mathrm{ }\text{failed}\mathrm{)}$ are also investigated.

Keywords:

• Crossed product
• group identity
• group ring
• non-matrix identity

2020 Mathematics Subject Classification:

• 16K40
• 16R40
• 16S34
• 16S35

Acknowledgments

The author would like to thank Professor Passman for reading the primary version of the manuscript and encouragement.

Additional information

Funding

The author was supported by Kharazmi University under Grant No. D/3802.