  Let $G$ be a finite group and $K$ a splitting field of $G$ of characteristic
$p>0$. Denote by $KG$ the group algebra of $G$ over $K$ and $Z(KG)$ the center
of $KG$. Let $V_G$ be the $K$-subspace of trace-zero elements of $KG$. We give
some numerical sufficient and necessary conditions for $V_G$ and $V_G\cap
Z(KG)$, respectively, to be Mathieu subspaces of $KG$ in terms of the degrees
of irreducible representations of $G$ over $K$. The same numerical conditions
also characterize the finite groups $G$ that $KG$ has no nonzero trace-zero
idempotents and the finite groups $G$ that $KG$ has no nonzero central
trace-zero idempotents, respectively.
