2007/09/30 by Abraham Broer, Jianjun Chuai
Mathematics · #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Combinatorics #Discrete mathematics #Finite Group Theory Research #Finitely-generated abelian group #Group (periodic table) #Group ring #Invariant (physics) #Invariant theory #Mathematical physics #Mathematics #Physics #Pure mathematics #Quantum mechanics #Vector space #math.AC #msc:13A50
paper · pdf · doi:10.1112/plms/pdp044
36 pages, proofs of main theorems have been improved
arxiv created 2009/04/06 · openalex publication_date 2009/12/13 · arxiv updated 2014/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let the finite group G act linearly on the vector space V over the field k of arbitrary characteristic, and let H < G be a subgroup. The extension of invariant rings k[V]G ⊂ k[V]H is studied using modules of covariants. An example of our results is the following. Let W be the subgroup of G generated by the reflections in G. A classical theorem due to Serre says that if k[V] is a free k[V]G-module then G = W. We generalize this result as follows. If k[V]H is a free k[V]G-module, then G is generated by H and W. Furthermore, the invariant ring k[V]H ∩ W is free over k[V]W and is generated as an algebra by H-invariants and W-invariants.