Difference between revisions of "Zassenhaus's Lemma"
m (→Resources: typo) |
Hashtagmath (talk | contribs) m (→Resources) |
||
| (One intermediate revision by one other user not shown) | |||
| Line 14: | Line 14: | ||
<cmath> (\eta^{-1} \circ \eta)(H \cap K')= H' \cdot (H\cap K') </cmath> | <cmath> (\eta^{-1} \circ \eta)(H \cap K')= H' \cdot (H\cap K') </cmath> | ||
is a normal subgroup of | is a normal subgroup of | ||
| − | <cmath> (\eta^{-1} \circ \eta)(H \cap K) = H' \cdot (H \cap K | + | <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 | 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> | <cmath> (H \cap K) \cap \bigl(H' \cdot (H \cap K') \bigr) = (H' \cap K) \cdot (H \cap K') . </cmath> | ||
| Line 26: | Line 26: | ||
* [[Jordan-Hölder Theorem]] | * [[Jordan-Hölder Theorem]] | ||
| − | [[Category:Group theory]] | + | [[Category:Group theory]] [[Category: Theorems]] |
Latest revision as of 11:13, 9 April 2019
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
![\[\bigl(H' \cdot(H \cap K)\bigr) / \bigl(H' \cdot (H \cap K') \bigr)\]](http://latex.artofproblemsolving.com/5/b/9/5b9b5e902aac391aa79832f05b549f8bfc21b824.png)
and
![\[\bigl(K' \cdot(H \cap K) \bigr) / \bigl(K' \cdot (K \cap H') \bigr)\]](http://latex.artofproblemsolving.com/b/f/f/bff8d9a4c2160bcc5ef1ac8307956ca1780259ec.png)
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
![\[(\eta^{-1} \circ \eta)(H \cap K')= H' \cdot (H\cap K')\]](http://latex.artofproblemsolving.com/4/2/f/42f95366f016a09f745cea479668fe41a6383177.png)
is a normal subgroup of
![\[(\eta^{-1} \circ \eta)(H \cap K) = H' \cdot (H \cap K) .\]](http://latex.artofproblemsolving.com/2/8/d/28d0ed79c34da8ff633b0426b16717cca5ad84a1.png)
Now, let 
be the canonical homomorphism from to 
. Now, note that
![\[(H \cap K) \cap \bigl(H' \cdot (H \cap K') \bigr) = (H' \cap K) \cdot (H \cap K') .\]](http://latex.artofproblemsolving.com/2/9/f/29f3c65b62dcae50780015ab5b6617f9bbf343dd.png)
Thus by the group homomorphism theorems, groups 
and
![\[(H \cap K)/ \bigl( (H' \cap K) \cdot (H \cap K') \bigr)\]](http://latex.artofproblemsolving.com/e/0/a/e0a9e78bca4c31945c74ccdac621cffd86eb4e65.png)
are isomorphic. The lemma then follows from symmetry between and . 
