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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Zassenhaus's Lemma

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Proof

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