normality satisfies image condition - groupprops

Normality satisfies image conditionFrom Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., normalsubgroup) satisfying a subgroup metaproperty (i.e., image condition)View all subgroup metaproperty satisfactions | View all subgroup metapropertydissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgrouppropertiesGet more facts about normal subgroup |Get facts that use property satisfaction of normalsubgroup | Get facts that use property satisfaction of normal subgroup|Get more facts aboutimage condition

Contents1 Statement

1.1 Property-theoretic statement1.2 Statement with symbols

2 Generalizations3 Proof

StatementProperty-theoretic statementThe subgroup property of being normal satisfies the image condition: the image of a normal subgroup underany surjective homomorphism is also normal.

Statement with symbolsSuppose is a surjective homomorphism of groups, and is a normal subgroup of .Then, is normal in .

GeneralizationsThis result is part of a more general result called the fourth isomorphism theorem (also called the latticeisomorphism theorem or correspondence theorem).

ProofGiven: 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 .

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