A note on the isomorphism problem for \(SK[G]\) (Q2758047)

Suppose \(G\) is an Abelian \(p\)-group, \(K\) is a field of characteristic different from \(p\) and \(KG\) is the group algebra of \(G\) over \(K\). We let \(V(KG)\) denote the group of normalized units (i.e. of augmentation 1) in \(KG\) and \(S(KG)\) the \(p\)-component of \(V(KG)\). Suppose \(\mu_p\) is the group of all \(p^n\)-th roots of 1. The field \(K\) is said to be of the first kind with respect to \(p\), if \([K(\mu_p):K]=\infty\), otherwise it is of the second kind with respect to \(p\). Let \(k\) be an uncountable regular cardinal and let \({\mathcal A}(k)\) be the set of the Abelian \(p\)-groups of cardinality \(k\). The main result of the paper is the following. Let \(K\) be a field of the first kind with respect to \(p\). When \(G\) runs over \({\mathcal A}(k)\) then there are \(2^k\) non-isomorphic groups \(S(KG)\).