  Let $G$ be a finite group and $N$ be a normal subgroup of $G$. Let
$J=J(F[N])$ denote the Jacboson radical of $F[N]$ and $I={\rm Ann}(J)=\{\alpha
\in F[G]|J\alpha =0\}$. We have another algebra $F[G]/I$. We study the
decomposition of Cartan matrix of $F[G]$ according to $F[G/N]$ and $F[G]/I$.
This decomposition establishs some connections between Cartan invariants and
chief composition factors of $G$. We find that existing zero-defect $p$-block
in $N$ depends on the properties of $I$ in $F[G]$ or Cartan invariants. When we
consider the Cartan invariants for a block algebra $B$ of $G$, the
decomposition is related to what kind of blocks in $N$ covered by $B$. We
mainly consider a block $B$ of $G$ which covers a block $b$ of $N$ with
$l(b)=1$. In two cases, we prove Willems' conjecture holds for these blocks,
which covers some true cases by Holm and Willems. Furthermore We give an
affirmative answer to a question by Holm and Willems in our cases. Some other
results about Cartan invariants are presented in our paper.
