The Schreier refinement theorem is a result in group theory that deals with the relationship between subgroups and normal series of a group. It provides criteria for refining a normal series of groups, allowing for more structured decompositions of groups into simpler components. The theorem is primarily used in the study of group extensions and solvable groups.
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Any two finite subnormal series have refinements with the same successive factors up to ordering and group isomorphism. For descending chains and , insert between and and insert in the other chain. The Zassenhaus butterfly lemma identifies the factor indexed by in the first refinement with the factor indexed by in the second. If both chains are composition series, their simple factors admit no nontrivial refinement, giving the Jordan–Hölder theorem.