Finite groups of OTP projective representation type

Leonid F. Barannyk¹

¹ Institute of Mathematics Pomeranian University of Słupsk Arciszewskiego 22b 76-200 Słupsk, Poland

Colloquium Mathematicum, Tome 126 (2012) no. 1, pp. 35-51

Résumé

Let $K$ be a field of characteristic $p>0$, $K^*$ the multiplicative group of $K$ and $G=G_p\times B$ a finite group, where $G_p$ is a $p$-group and $B$ is a $p'$-group. Denote by $K^\lambda G$ a twisted group algebra of $G$ over $K$ with a $2$-cocycle $\lambda \in Z^2(G,K^*)$. We give necessary and sufficient conditions for $G$ to be of OTP projective $K$-representation type, in the sense that there exists a cocycle $\lambda \in Z^2(G,K^*)$ such that every indecomposable $K^\lambda G$-module is isomorphic to the outer tensor product $V\mathbin {\#} W$ of an indecomposable $K^\lambda G_p$-module $V$ and a simple $K^\lambda B$-module $W$. We also exhibit finite groups $G=G_p\times B$ such that, for any $\lambda \in Z^2(G,K^*)$, every indecomposable $K^\lambda G$-module satisfies this condition.

Détail

DOI : 10.4064/cm126-1-2

Keywords: field characteristic * multiplicative group times finite group where p group p group denote lambda twisted group algebra cocycle lambda * necessary sufficient conditions otp projective k representation type sense there exists cocycle lambda * every indecomposable lambda g module isomorphic outer tensor product mathbin indecomposable lambda p module simple lambda b module exhibit finite groups times lambda * every indecomposable lambda g module satisfies condition