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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Statement with symbols
Suppose is a surjective homomorphism of groups, and is a normal subgroup of . Then, is normal in .
This result is part of a more general result called the fourth isomorphism theorem (also called the lattice isomorphism theorem or correspondence theorem).
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 .