Schreier Refinement Theorem
Any two subnormal series admit equivalent refinements with isomorphic factors
Schreier Refinement Theorem. Let be a group. Consider two finite subnormal series
Then the two series have refinements, obtained by inserting intermediate subgroups, whose successive factor groups are pairwise isomorphic after a permutation. In particular, the refined series have the same length.
Remarks
Schreier refinement is the main structural comparison tool for normal series. It is the standard input for the Jordan–Hölder theorem.