By P. Eymard, J. Faraut, G. Schiffmann, R. Takahashi
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B) Let y ∈ Ah be any homogeneous element of A. Then yzg = yz = zy = zg y. Since H is abelian and yzg is in Ahg = Agh , and zg y is in Agh , we have yzg = zg y for all g ∈ H. Hence zg is in Z(A) as desired. We will need the following important result about central units in integral group rings. 2. 3] If α is a central unit of finite order in ZG, then α belongs to ±Z(G). 3. Assume ZG is graded by a finite abelian 2-group K. If g ∈ Z(G) is of finite order then g is homogeneous. Proof. We proceed by induction on |K|.
C. K. Sehgal, An Introduction to Group Rings, Algebras and Applications, Kluwer, 2002. K. Sehgal, Topics in Group Rings, Marcel Dekker, 1978. K. Sehgal, Units in Integral Group Rings, Longmans, 1999. at Dedicated to C. Polcino Milies, S. Sehgal, and S. Sidki. Abstract. Defining a profinite graph to be a projective limit of an inverse system of finite graphs can lead to different objects depending upon which of the (usually) equivalent concepts of a finite graph is employed. 1. Categories of (profinite) graphs Graphs can be oriented - each edge has an origin and a terminal vertex or nonoriented.
4. (a) The functor L : H M → G MG/H given by LN (U ) = H OG (π −1 (U )) ⊗ N for a rational H-module N and U open subset of G/H, is an equivalence of categories, whose inverse is given by taking the fiber at the identity: F → FeH , F ∈ G MG/H . (b) L is a module functor. If K is a subgroup of H then the following diagram commutes: HM L −−−−→ Res KM G MG/H π ∗ L −−−−→ G MG/K , where π : G/H → G/K is the canonical projection. Proof. (a) is [CPS, Th. 7]. The proof of (b) is straightforward. 5.
Analyse Harmonique sur les Groupes de Lie by P. Eymard, J. Faraut, G. Schiffmann, R. Takahashi