On twisted group algebras of OTP representation type over the ring of $p$-adic integers
Volume 143 / 2016
Colloquium Mathematicum 143 (2016), 209-235
MSC: Primary 16G60; Secondary 20C20, 20C25.
DOI: 10.4064/cm6700-1-2016
Published online: 12 January 2016
Abstract
Let $\hat{\mathbb{Z}}_p$ be the ring of $p$-adic integers, $U(\hat{\mathbb{Z}}_p)$ the unit group of $\hat{\mathbb{Z}}_p$ and $G=G_p\times B$ a finite group, where $G_p$ is a $p$-group and $B$ is a $p’$-group. Denote by $\hat{\mathbb{Z}}_p^\lambda G$ the twisted group algebra of $G$ over $\hat{\mathbb{Z}}_p$ with a $2$-cocycle $\lambda\in Z^2(G,U(\hat{\mathbb{Z}}_p))$. We give necessary and sufficient conditions for $\hat{\mathbb{Z}}_p^\lambda G$ to be of OTP representation type, in the sense that every indecomposable $\hat{\mathbb{Z}}_p^\lambda G$-module is isomorphic to the outer tensor product $V\mathbin{\#} W$ of an indecomposable $\hat{\mathbb{Z}}_p^\lambda G_p$-module $V$ and an irreducible $\hat{\mathbb{Z}}_p^\lambda B$-module $W$.
