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
Property-theoretic 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.