Direct sums and products of isomorphic Abelian groups (Q1093017)

The result of this note is the following: If G is a reduced abelian group, I, J are infinite sets and either \(| I|\) or \(| G|\) is non-measurable, then \(G^ I=\oplus_ JA\), for some subgroup A of G if and only if \(G=B\oplus C\), where \(B^ I\cong \oplus_ JT\), for some bounded subgroup T and \(C^ I\cong \oplus_ JC\cong C^ k\), for some positive integer k. An example of a reduced unbounded group G, such that \(G\cong G^ I\cong \oplus_ JG\) (I, J infinite sets) is obtained. If I is a set of measurable cardinality, then, for every group G, \(G^ I=L\oplus M\), where \(L\cong G\), \(M\cong G^ I\) and \(\oplus_ IG\) is a genuine subgroup of M.