Let $G$ be a finite group and $H$ a subgroup. Denote by $D(G;H)$ (or $D\left(G\right)$) the crossed product of $C\left(G\right)$ and $ℂH$ (or $ℂG$) with respect to the adjoint action of the latter on the former. Consider the algebra $\langle D\left(G\right),e\rangle$ generated by $D\left(G\right)$ and $e$, where we regard $E$ as an idempotent operator $e$ on $D\left(G\right)$ for a certain conditional expectation $E$ of $D\left(G\right)$ onto $D(G;H)$. Let us call $\langle D\left(G\right),e\rangle$ the basic construction from the conditional expectation $E:D\left(G\right)\to D(G;H)$. The paper constructs a crossed product algebra $C(G/H\times G)⋊ℂG$, and proves that there is an algebra isomorphism between $\langle D\left(G\right),e\rangle$ and $C(G/H\times G)⋊ℂG$.

Let S be a commutative local ring of characteristic p, which is not a field, S* the multiplicative group of S, W a subgroup of S*, G a finite p-group, and $S^{\lambda }G$ a twisted group ring of the group G and of the ring S with a 2-cocycle λ ∈ Z²(G,S*). Denote by $In{d}_m\left(S^{\lambda }G\right)$ the set of isomorphism classes of indecomposable $S^{\lambda }G$-modules of S-rank m. We exhibit rings $S^{\lambda }G$ for which there exists a function $f_{\lambda }:\mathbb{N}\to \mathbb{N}$ such that $f_{\lambda }\left(n\right)\ge n$ and $In{d}_{f_{\lambda }\left(n\right)}\left(S^{\lambda }G\right)$ is an infinite set for every natural n > 1. In special cases $f_{\lambda }\left(\mathbb{N}\right)$ contains every natural number m >...