Finite groups of OTP projective representation type over a complete discrete valuation domain of positive characteristic

Leonid F. Barannyk ¹   ; Dariusz Klein ¹

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

Colloquium Mathematicum, Tome 129 (2012) no. 2, pp. 173-187

Résumé

Let $S$ be a commutative complete discrete valuation domain of positive characteristic $p$, $S^*$ the unit group of $S$, $\varOmega $ a subgroup of $S^*$ and $G=G_p\times B$ a finite group, where $G_p$ is a $p$-group and $B$ is a $p'$-group. Denote by $S^\lambda G$ the twisted group algebra of $G$ over $S$ with a $2$-cocycle $\lambda \in Z^2(G,S^*)$. For $\varOmega $ satisfying a specific condition, we give necessary and sufficient conditions for $G$ to be of OTP projective $(S,\varOmega )$-representation type, in the sense that there exists a cocycle $\lambda \in Z^2(G,\varOmega )$ such that every indecomposable $S^\lambda G$-module is isomorphic to the outer tensor product $V\mathbin {\#} W$ of an indecomposable $S^\lambda G_p$-module $V$ and an irreducible $S^\lambda B$-module $W$.

Détail

DOI : 10.4064/cm129-2-2

Keywords: commutative complete discrete valuation domain positive characteristic * unit group varomega subgroup * times finite group where p group p group denote lambda twisted group algebra cocycle lambda * varomega satisfying specific condition necessary sufficient conditions otp projective varomega representation type sense there exists cocycle lambda varomega every indecomposable lambda g module isomorphic outer tensor product mathbin indecomposable lambda p module irreducible lambda b module