  Let $G$ be a group, $R$ an integral domain, and $V_G$ the subspace of the
group algebra $R[G]$ consisting of all the elements of $R[G]$ whose coefficient
of the identity element $1_G$ of $G$ is equal to zero. Motivated by the Mathieu
conjecture [M], the Duistermaat-van der Kallen theorem [DK], and also by recent
studies on the notion of Mathieu subspaces introduced in [Z4] and [Z6], we show
that for finite groups $G$, $V_G$ under certain conditions also forms a Mathieu
subspace of the group algebra $R[G]$. We also show that for the free abelian
groups $G=\Bbb Z^n$ $(n\ge 1)$ and any integral domain $R$ of positive
characteristic, $V_G$ fails to be a Mathieu subspace of $R[G]$, which is
equivalent to saying that the Duistermaat-van der Kallen theorem [DK] cannot be
generalized to any field or integral domain of positive characteristic.
