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An analogue of the Duistermaat-van der Kallen theorem for group algebras

Wenhua Zhao, Roel Willems (2012)

Open Mathematics

Let G be a group, R an integral domain, and V G the R-subspace of the group algebra R[G] consisting of all the elements of R[G] whose coefficient of the identity element 1G of G is equal to zero. Motivated by the Mathieu conjecture [Mathieu O., Some conjectures about invariant theory and their applications, In: Algèbre non Commutative, Groupes Quantiques et Invariants, Reims, June 26–30, 1995, Sémin. Congr., 2, Société Mathématique de France, Paris, 1997, 263–279], the Duistermaat-van der Kallen...

Augmentation quotients for Burnside rings of generalized dihedral groups

Shan Chang (2016)

Czechoslovak Mathematical Journal

Let H be a finite abelian group of odd order, 𝒟 be its generalized dihedral group, i.e., the semidirect product of C 2 acting on H by inverting elements, where C 2 is the cyclic group of order two. Let Ω ( 𝒟 ) be the Burnside ring of 𝒟 , Δ ( 𝒟 ) be the augmentation ideal of Ω ( 𝒟 ) . Denote by Δ n ( 𝒟 ) and Q n ( 𝒟 ) the n th power of Δ ( 𝒟 ) and the n th consecutive quotient group Δ n ( 𝒟 ) / Δ n + 1 ( 𝒟 ) , respectively. This paper provides an explicit -basis for Δ n ( 𝒟 ) and determines the isomorphism class of Q n ( 𝒟 ) for each positive integer n .

Brauer relations in finite groups

Alex Bartel, Tim Dokchitser (2015)

Journal of the European Mathematical Society

If G is a non-cyclic finite group, non-isomorphic G -sets X , Y may give rise to isomorphic permutation representations [ X ] [ Y ] . Equivalently, the map from the Burnside ring to the rational representation ring of G has a kernel. Its elements are called Brauer relations, and the purpose of this paper is to classify them in all finite groups, extending the Tornehave–Bouc classification in the case of p -groups.

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