The simplest example of two non-isomorphic groups with the same character tables is provided by the non-abelian groups of order p3, p ≠ 2. Let G1 be the one of exponent p and let G2 be the other. If Q denotes the field of rational numbers, then Berman (2) has shown that QG1 ≈ QG2, where QGi denotes the rational group algebra. In this note we shall show that the corresponding statement is false for ZGi where Z is the ring of rational integers. More explicitly we shall show that ZG1 does not contain a unit of order p2 so that it is impossible to embed ZG2 in ZG1.