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Let \(G\) be a finite group and \(R(G)\) be its character ring, i.e. the set of
\(\mathbb{Z}\) -
linear combinations of complex characters of \(G.\) Then \(R(G)\) is a \(\mathbb{Z}\)-order in \(\mathbb{Q} \otimes R(G).\) A natural strategy to study algebraic properties of the character
ring is to consider the maximal order in \(\mathbb{Q} \otimes R(G).\) Doing this we will see that
it is not diļ¬cult to determine the unit group and the Brauer group of \(R(G).\)
However, in general it seems to be hard to decide whether the representation
type of \(R(G)\) is finite. We will give some necessary and sufficient conditions
for the finiteness of the representation type of \(R(G).\) These yield the answer
to this problem in a number of cases.
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