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 19:38, 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
.