View source for 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 <math>G</math> be a [[group]]; let <math>H</math>, <math>H'</math>, <math>K</math>, <math>K'</math> be [[subgroup]]s of <math>G</math> such that <math>H'</math> is a [[normal subgroup]] of <math>H</math> and <math>K'</math> is a normal subgroup of <math>K</math>.  Then <math>H'\cdot(H \cap K')</math> is a normal subgroup of <math>H' \cdot (H \cap K)</math>; likewise, <math>K' \cdot (K \cap H')</math> is a normal subgroup of <math>K' \cdot (H \cap K)</math>; furthermore, the [[quotient group]]s
<cmath> \bigl(H' \cdot(H \cap K)\bigr) / \bigl(H' \cdot (H \cap K') \bigr) </cmath>
and
<cmath> \bigl(K' \cdot(H \cap K) \bigr) / \bigl(K' \cdot (K \cap H') \bigr) </cmath>
are [[isomorphic]].

== Proof ==

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

== Resources ==

* [[Schreier's Theorem]]
* [[Jordan-Hölder Theorem]]

[[Category:Group theory]]