Normality satisfies inverse image condition
From Groupprops
This article gives the statement, and possibly proof, of a subgroup property satisfying a subgroup metaproperty
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Contents
Statement
Propertytheoretic statement
The subgroup property of being normal satisfies the subgroup metaproperty called the inverse image condition: the inverse image of a normal subgroup, under a homomorphism, is normal.
Statement with symbols
Let be a homomorphism of groups, and be a normal subgroup of . Then, is a normal subgroup of .
Related facts
Proof
Given: , a homomorphism of groups, and is a normal subgroup of
To prove: is normal in
Proof: Pick and . We need to show that .
By the fact that is a homomorphism:
Since , , and since is normal in , the right side of the above equation is in . Hence, , so , as required.