Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn partial augmentations of elements in integral group rings
100
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Inner relations are derived between partial augmentations of certain elements (units or idempotents) in group rings.
rate research
Read More
We study partial homology and cohomology from ring theoretic point of view via the partial group algebra $mathbb{K}_{par}G$. In particular, we link the partial homology and cohomology of a group $G$ with coefficients in an irreducible (resp. indecomposable) $mathbb{K}_{par}G$-module with the ordinary homology and cohomology groups of $G$ with in general non-trivial coefficients. Furthermore, we compare the standard cohomological dimension $cd_{ mathbb{K}}(G)$ (over a field $mathbb{K}$) with the partial cohomological dimension $cd_{ mathbb{K}}^{par}(G)$ (over $mathbb{K}$) and show that $cd_{ mathbb{K}}^{par}(G) geq cd_{ mathbb{K}}(G)$ and that there is equality for $G = mathbb{Z}$.
Let $R$ be a commutative Noetherian ring with unit. We classify the characters of the group $mathrm{EL}_d (R)$ provided that $d$ is greater than the stable range of the ring $R$. It follows that every character of $mathrm{EL}_d (R)$ is induced from a finite dimensional representation. Towards our main result we classify $mathrm{EL}_d (R)$-invariant probability measures on the Pontryagin dual group of $R^d$.
For modules over group rings we introduce the following numerical parameter. We say that a module A over a ring R has finite r-generator property if each f.g. (finitely generated) R-submodule of A can be generated exactly by r elements and there exists a f.g. R-submodule D of A, which has a minimal generating subset, consisting exactly of r elements. Let FG be the group algebra of a finite group G over a field F. In the present paper modules over the algebra FG having finite generator property are described.
A long standing problem, which has its roots in low-dimensional homotopy theory, is to classify all finite groups $G$ for which the integral group ring $mathbb{Z}G$ has stably free cancellation (SFC). We extend results of R. G. Swan by giving a condition for SFC and use this to show that $mathbb{Z}G$ has SFC provided at most one copy of the quaternions $mathbb{H}$ occurs in the Wedderburn decomposition of the real group ring $mathbb{R}G$. This generalises the Eichler condition in the case of integral group rings.
This paper is a new contribution to the study of regular subgroups of the affine group $AGL_n(F)$, for any field $F$. In particular we associate to any partition $lambda eq (1^{n+1})$ of $n+1$ abelian regular subgroups in such a way that different partitions define non-conjugate subgroups. Moreover, we classify the regular subgroups of certain natural types for $nleq 4$. Our classification is equivalent to the classification of split local algebras of dimension $n+1$ over $F$. Our methods, based on classical results of linear algebra, are computer free.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
