Subnormal and normal series, factors, refinements, and equivalence

A subnormal series of a group $G$ is a finite chain $G=G_0⊵G_1⊵\cdots ⊵G_n=1$ in which $G_{i+1}⊴G_i$ for every $0\le i<n$ (Normal subgroup: invariance under conjugation). Its factors are the quotient groups $G_i\mathrm{/}{G}_{i+1}$. The case $n=0$ is allowed and is the unique subnormal series of the trivial group.

A normal series is a subnormal series in which every $G_i$ is normal in $G$. Thus “normal series” is stronger than “subnormal series” here.

A subnormal series $G=H_0⊵\cdots ⊵H_m=1$ is a refinement of the displayed series if the $G_i$ occur among the $H_j$ in the same order. Repeated adjacent terms may be deleted without changing the nontrivial factors. Two subnormal series are equivalent if, after deleting repeated adjacent terms, their factors can be paired by a permutation so that paired factors are isomorphic (Group isomorphisms, automorphisms and the set $\mathrm{Aut}(G)$).