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Concept# Schur–Zassenhaus theorem

Summary

The Schur–Zassenhaus theorem is a theorem in group theory which states that if G is a finite group, and N is a normal subgroup whose order is coprime to the order of the quotient group G/N, then G is a semidirect product (or split extension) of N and G/N. An alternative statement of the theorem is that any normal Hall subgroup N of a finite group G has a complement in G. Moreover if either N or G/N is solvable then the Schur–Zassenhaus theorem also states that all complements of N in G are conjugate. The assumption that either N or G/N is solvable can be dropped as it is always satisfied, but all known proofs of this require the use of the much harder Feit–Thompson theorem.
The Schur–Zassenhaus theorem at least partially answers the question: "In a composition series, how can we classify gro

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