Normality is quotient-transitive

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This article gives the statement, and possibly proof, of a subgroup property satisfying a subgroup metaproperty
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Statement

Symbolic statement

Suppose H is a normal subgroup of G and K is a subgroup of G containing H, such that K/H is normal in G/H. Then K is also normal in G.

Related facts

Generalization and other particular cases

A general version is:

Quotient-balanced implies quotient-transitive

Other particular cases are:

Related facts about normality

Proof

Hands-on proof

Pick g \in G. Then the map c_g:x \mapsto gxg^{-1} is an inner automorphism of G, hence sends H to itself and induces an automorphism of G/H. More importantly, the induced automorphism on G/H is also an inner automorphism, by the image of g under the quotient map G \to G/H. Since K/H is normal in G/H, the induced map on G/H preserves the subgroup K/H. Hence c_g sends K to K.

Property-theoretic proof