Normal series of a group

A normal series is a chain $1=G_0\le G_1\le \cdots \le G_r=G$ whose terms are all normal subgroups of the ambient group. A subnormal series only requires each term normal in the next. Confusing these conditions would incorrectly require every term of a composition series to be normal in the full group.

A chief series is a strict normal series of a group with no proper normal-series refinement. Its factors are minimal normal subgroups of the appropriate quotient groups. A chief factor of a finite soluble group is elementary abelian, whereas a composition factor of a finite soluble group is cyclic of prime order. A chief factor can therefore have order larger than a prime.

A hypothetical finite chief series in $\mathbb{Z}$ has a first nonzero term $n\mathbb{Z}$. The subgroup $2n\mathbb{Z}$ is normal in $\mathbb{Z}$ and strictly between zero and that term. This contradicts the defining maximality condition for the first chief factor. Finite groups always admit chief series, by choosing a maximal chain of their normal subgroups; this example shows why finiteness matters.

A chief factor of $G$ is a quotient $N/M$ with $M,N$ normal in $G$ and no $G$-normal subgroup strictly between them. Equivalently, $N/M$ is a minimal normal subgroup of $G/M$. Such factors are the successive factors of a chief series, and are direct products of isomorphic simple groups when $G$ is finite.