Schreier Refinement Theorem

Schreier Refinement Theorem. Let $G$ be a group. Consider two finite subnormal series (normal series)

where each inclusion is normal in the previous term. Then there exist refinements of these series (obtained by inserting additional intermediate subgroups) such that the refined series have the same length and their successive factor groups are pairwise isomorphic up to a permutation. Each factor group is a quotient group of the form $A/B$ with $B \trianglelefteq A$.