= Schreier refinement theorem
{c}
Any two finite <subnormal series> have refinements with the same successive factors up to ordering and <group isomorphism>. For descending chains $A_i$ and $B_j$, insert $A_{i+1}(A_i\cap B_j)$ between $A_i$ and $A_{i+1}$ and insert $B_{j+1}(B_j\cap A_i)$ in the other chain. The <Zassenhaus butterfly lemma> identifies the factor indexed by $(i,j)$ in the first refinement with the factor indexed by $(j,i)$ in the second. If both chains are <composition series>, their simple factors admit no nontrivial refinement, giving the <Jordan–Hölder theorem>.
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