Schur–Zassenhaus Theorem

Schur–Zassenhaus Theorem. Let $G$ be a finite group and let $N \trianglelefteq G$ be a normal subgroup that is a Hall subgroup (equivalently, $\gcd(|N|,[G:N])=1$). Then:

1. (Existence of complements) There exists a subgroup $H \le G$ such that $G = NH$ and $N \cap H = \{e\}$. Equivalently, $G$ is a semidirect product $G \cong N \rtimes H$.
2. (Conjugacy of complements) If $H_1$ and $H_2$ are two such complements, then $H_1$ and $H_2$ are conjugate in $G$ (they lie in the same orbit under the conjugation action ).

This theorem is often phrased as: a short exact sequence with coprime kernel and quotient splits . It is a central tool for analyzing group structure when orders factor into coprime parts.

Proof sketch. One proves existence by induction on $|G|$, using coprimality to force fixed points in appropriate actions and to build a complement step-by-step. Conjugacy of complements is obtained by a similar induction argument, comparing two complements via their actions on $N$ and exploiting the coprime condition to solve a conjugacy equation.