Let $G$ be an abelian group, $R$ a commutative ring of prime characteristic $p$ with identity and $R_{t}G$ a commutative twisted group ring of $G$ over $R$. Suppose $p$ is a fixed prime, $G_p$ and $S\left(R_{t}G\right)$ are the $p$-components of $G$ and of the unit group $U\left(R_{t}G\right)$ of $R_{t}G$, respectively. Let $R^{*}$ be the multiplicative group of $R$ and let $f_{\alpha }\left(S\right)$ be the $\alpha$-th Ulm-Kaplansky invariant of $S\left(R_{t}G\right)$ where $\alpha$ is any ordinal. In the paper the invariants $f_n\left(S\right)$, $n\in \mathbb{N}\cup \left\{0\right\}$, are calculated, provided $G_p=1$. Further, a commutative ring $R$ with identity of prime characteristic $p$ is said...

Let G be a noncyclic abelian p-group and K be an infinite field of finite characteristic p. For every 2-cocycle λ ∈ Z²(G,K*) such that the twisted group algebra $K^{\lambda }G$ is of infinite representation type, we find natural numbers d for which G has infinitely many faithful absolutely indecomposable λ-representations over K of dimension d.

Let G be a finite group, F a field of characteristic p with p||G|, and $F^{\lambda }G$ the twisted group algebra of the group G and the field F with a 2-cocycle λ ∈ Z²(G,F*). We give necessary and sufficient conditions for $F^{\lambda }G$ to be of finite representation type. We also introduce the concept of projective F-representation type for the group G (finite, infinite, mixed) and we exhibit finite groups of each type.

Let Λ = (S/R,α) be a local weak crossed product order in the crossed product algebra A = (L/K,α) with integral cocycle, and $H=\sigma \in Gal(L/K)|\alpha (\sigma ,{\sigma }^{-1})\in S*$ the inertial group of α, for S* the group of units of S. We give a condition for the first ramification group of L/K to be a subgroup of H. Moreover we describe the Jacobson radical of Λ without restriction on the ramification of L/K.

Our aim is to determine necessary and sufficient conditions for a finite nilpotent group to have a faithful irreducible projective representation over a field of characteristic p ≥ 0.

Assume that S is a commutative complete discrete valuation domain of characteristic p, S* is the unit group of S and $G=G_p\times B$ is 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 λ ∈ Z²(G,S*). We give necessary and sufficient conditions for $S^{\lambda }G$ to be of OTP representation type, in the sense that every indecomposable $S^{\lambda }G$-module is isomorphic to the outer tensor product V W of an indecomposable $S^{\lambda }{G}_p$-module V and an irreducible $S^{\lambda }B$-module...