Normality satisfies image condition
This article gives the statement, and possibly proof, of a subgroup property (i.e., normal subgroup) satisfying a subgroup metaproperty (i.e., image condition)
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Contents
Statement
Property-theoretic statement
The subgroup property of being normal satisfies the image condition: the image of a normal subgroup under any surjective homomorphism is also normal.
Statement with symbols
Suppose is a surjective homomorphism of groups, and
is a normal subgroup of
. Then,
is normal in
.
Generalizations
This result is part of a more general result called the fourth isomorphism theorem (also called the lattice isomorphism theorem or correspondence theorem).
Proof
Given: is a surjective homomorphism of groups, and
is a normal subgroup of
To prove: is normal in
Proof: Pick and
. We need to show that
.
Since , there exists
such that
. Further, since
is surjective, there exists
such that
. Then:
(where the second step uses the fact that is a homomorphism).
Now, since is normal in
,
, and hence
, showing that
.