Difference between revisions of "Zassenhaus's Lemma"
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<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> |
Revision as of 18:38, 23 April 2009
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 and 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 is a normal subgroup of Now, let be the canonical homomorphism from to . Now, note that Thus by the group homomorphism theorems, groups and are isomorphic. The lemma then follows from symmetry between and .