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Cokernels of Homomorphisms from Burnside Rings to Inverse Limits

  Published:2016-11-29
 Printed: Mar 2017
  • Masaharu Morimoto,
    Graduate School of Natural Science and Technology, Okayama University, Tsushimanaka 3-1-1, Kitaku, Okayama, 700-8530 Japan
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Abstract

Let $G$ be a finite group and let $A(G)$ denote the Burnside ring of $G$. Then an inverse limit $L(G)$ of the groups $A(H)$ for proper subgroups $H$ of $G$ and a homomorphism ${\operatorname{res}}$ from $A(G)$ to $L(G)$ are obtained in a natural way. Let $Q(G)$ denote the cokernel of ${\operatorname{res}}$. For a prime $p$, let $N(p)$ be the minimal normal subgroup of $G$ such that the order of $G/N(p)$ is a power of $p$, possibly $1$. In this paper we prove that $Q(G)$ is isomorphic to the cartesian product of the groups $Q(G/N(p))$, where $p$ ranges over the primes dividing the order of $G$.
Keywords: Burnside ring, inverse limit, finite group Burnside ring, inverse limit, finite group
MSC Classifications: 19A22, 57S17 show english descriptions Frobenius induction, Burnside and representation rings
Finite transformation groups
19A22 - Frobenius induction, Burnside and representation rings
57S17 - Finite transformation groups
 

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