# Zassenhaus's Lemma

**Zassenhaus's Lemma** is a result in group theory. Hans Zassenhaus published his proof of the lemma in 1934 to provide a more elegant proof of Schreier's Theorem. He was a doctorate student under Emil Artin at the time. In this article, group operation is written multiplicatively.

## Statement

Let be a group; let , , , be subgroups of such that is a normal subgroup of and is a normal subgroup of . Then is a normal subgroup of ; likewise, is a normal subgroup of ; furthermore, the quotient groups and are isomorphic.

## Proof

We first note that is a subgroup of . Let be the canonical homomorphism from to . Then , so this indeed a group. Also, note that is a normal subgroup of . Hence is a normal subgroup of Now, let be the canonical homomorphism from to . Now, note that Thus by the group homomorphism theorems, groups and are isomorphic. The lemma then follows from symmetry between and .