Difference between revisions of "Zassenhaus's Lemma"
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Latest revision as of 12:13, 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
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
.