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 $G$ be a group; let $H$, $H'$, $K$, $K'$ be subgroups of $G$ such that $H'$ is a normal subgroup of $H$ and $K'$ is a normal subgroup of $K$. Then $H'\cdot(H \cap K')$ is a normal subgroup of $H' \cdot (H \cap K)$; likewise, $K' \cdot (K \cap H')$ is a normal subgroup of $K' \cdot (H \cap K)$; furthermore, the quotient groups \[\bigl(H' \cdot(H \cap K)\bigr) / \bigl(H' \cdot (H \cap K') \bigr)\] and \[\bigl(K' \cdot(H \cap K) \bigr) / \bigl(K' \cdot (K \cap H') \bigr)\] are isomorphic.

Proof

We first note that $H \cap K$ is a subgroup of $H$. Let $\eta$ be the canonical homomorphism from $H$ to $H/H'$. Then $(\eta^{-1} \circ \eta)(H\cap K) = H' \cdot (H \cap K)$, so this indeed a group. Also, note that $H \cap K'$ is a normal subgroup of $H \cap K$. Hence \[(\eta^{-1} \circ \eta)(H \cap K')= H' \cdot (H\cap K')\] is a normal subgroup of \[(\eta^{-1} \circ \eta)(H \cap K) = H' \cdot (H \cap K) .\] Now, let $\lambda$ be the canonical homomorphism from $H' \cdot (H \cap K)$ to $\bigl(H' \cdot (H \cap K) \bigr)/ \bigl( H' \cdot (H \cap K') \bigr)$. Now, note that \[(H \cap K) \cap \bigl(H' \cdot (H \cap K') \bigr) = (H' \cap K) \cdot (H \cap K') .\] Thus by the group homomorphism theorems, groups $\bigl( H' \cdot (H \cap K) \bigr) / \bigl( H' \cdot (H \cap K') \bigr)$ and \[(H \cap K)/ \bigl( (H' \cap K) \cdot (H \cap K') \bigr)\] are isomorphic. The lemma then follows from symmetry between $H$ and $K$. $\blacksquare$

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