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Cokernels of Homomorphisms from Burnside Rings to Inverse Limits

Published online by Cambridge University Press:  20 November 2018

Masaharu Morimoto
Masaharu Morimoto*
Affiliation:
Graduate School of Natural Science and Technology, Okayama University, Tsushimanaka 3-1-1, Kitaku, Okayama, 700-8530Japan e-mail: morimoto@ems.okayama-u.ac.jp
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Abstract

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Let $G$ be a finite group and let $A\left( G \right)$ denote the Burnside ring of $G$. Then an inverse limit $L\left( G \right)$ of the groups $A\left( H \right)$ for proper subgroups $H$ of $G$ and a homomorphism res from $A\left( G \right)$ to $L\left( G \right)$ are obtained in a natural way. Let $Q\left( G \right)$ denote the cokernel of res. For a prime $p$, let $N\left( p \right)$ be the minimal normal subgroup of $G$ such that the order of ${G}/{N}\;\left( p \right)$ is a power of $p$, possibly 1. In this paper we prove that $Q\left( G \right)$ is isomorphic to the cartesian product of the groups $Q\left( {G}/{N\left( p \right)}\; \right)$, where $p$ ranges over the primes dividing the order of $G$.

Keywords

19A2257S17Burnside ringinverse limitfinite group
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 60 , Issue 1 , 01 March 2017 , pp. 165 - 172
Copyright
Copyright © Canadian Mathematical Society 2017

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