Schreier refinement theorem

In mathematics, the Schreier refinement theorem of group theory states that any two subnormal series of subgroups of a given group have equivalent refinements, where two series are equivalent if there is a bijection between their factor groups that sends each factor group to an isomorphic one.
The theorem is named after the Austrian mathematician Otto Schreier who proved it in 1928. It provides an elegant proof of the Jordan–Hölder theorem. It is often proved using the Zassenhaus lemma. gives a short proof by intersecting the terms in one subnormal series with those in the other series.
Example.
Consider formula_1, where formula_2 is the symmetric group of degree 3. There are subnormal series
formula_2 contains the normal subgroup formula_6. Hence these have refinements
with factor groups isomorphic to formula_8 and
with factor groups isomorphic to formula_10.