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On twisted group algebras of OTP representation type over the ring of $p$-adic integers

Volume 143 / 2016

Leonid F. Barannyk, Dariusz Klein 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 $W$.

Authors

  • Leonid F. BarannykInstitute of Mathematics
    Pomeranian University of Słupsk
    Arciszewskiego 22d
    76-200 Słupsk, Poland
    e-mail
  • Dariusz KleinInstitute of Mathematics
    Pomeranian University of Słupsk
    Arciszewskiego 22d
    76-200 Słupsk, Poland
    e-mail

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