Schur–Zassenhaus theorem

The Schur–Zassenhaus theorem is a theorem in group theory which states that if ${\displaystyle G}$ is a finite group, and ${\displaystyle N}$ is a normal subgroup whose order is coprime to the order of the quotient group ${\displaystyle G/N}$, then ${\displaystyle G}$ is a semidirect product (or split extension) of ${\displaystyle N}$ and ${\displaystyle G/N}$. An alternative statement of the theorem is that any normal Hall subgroup ${\displaystyle N}$ of a finite group ${\displaystyle G}$ has a complement in ${\displaystyle G}$. Moreover if either ${\displaystyle N}$ or ${\displaystyle G/N}$ is solvable then the Schur–Zassenhaus theorem also states that all complements of ${\displaystyle N}$ in G are conjugate. The assumption that either ${\displaystyle N}$ or ${\displaystyle G/N}$ 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.

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