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Zassenhaus's Lemma

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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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