Subnormal series

A subnormal series is a finite chain of subgroups in which each is a normal subgroup of the next. A member need not be normal in the whole group. Requiring the factors to be cyclic groups defines a polycyclic group; allowing finite factors as well defines a poly-(cyclic or finite) group.

Any two finite subnormal series have refinements with the same successive factors up to ordering and group isomorphism. For descending chains $A_i$ and $B_j$, insert $A_{i+1}(A_i\cap B_j)$ between $A_i$ and $A_{i+1}$ and insert $B_{j+1}(B_j\cap A_i)$ in the other chain. The Zassenhaus butterfly lemma identifies the factor indexed by $(i,j)$ in the first refinement with the factor indexed by $(j,i)$ in the second. If both chains are composition series, their simple factors admit no nontrivial refinement, giving the Jordan–Hölder theorem.

Let $A^{\prime }⊴A$ and $B^{\prime }⊴B$ be subgroups of one group. Put $C=A\cap B$ and $D=(A^{\prime }\cap B)(A\cap B^{\prime })$. Then $D⊴C$ and the first isomorphism theorem identifies bothwith $C/D$. Indeed $C$ maps onto either displayed quotient group, and its kernel in either case is $D$. This gives the matching factors used in the Schreier refinement theorem.

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 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.