## Every normal subgroup appears in a composition series

If $G$ is a finite group and $N \leq G$ is normal, prove that there is a composition series of $G$, one of whose terms is $N$.

Let $G$ be a finite group and $N \leq G$ normal. Let $1 = H_1 \vartriangleleft \cdots \vartriangleleft H_k = N$ and $1/N = K_1/N \vartriangleleft \cdots \vartriangleleft K_\ell/N = G/N$ be composition series for $N$ and $G/N$, respectively; these exist by the previous exercise. Then $1 = H_1 \vartriangleleft \cdots \vartriangleleft H_k = N = K_1 \vartriangleleft \cdots \vartriangleleft K_\ell = G$ is a composition series for $G$, one of whose terms is $N$.