Normality is quotient-transitive

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

 * Third isomorphism theorem is a statement describing the isomorphism types of the various quotient groups.

Generalization and other particular cases
A general version is:

Quotient-balanced implies quotient-transitive

Other particular cases are:


 * Characteristicity is quotient-transitive
 * Strict characteristicity is quotient-transitive
 * Full invariance is quotient-transitive

Related facts about normality

 * Normality is strongly join-closed
 * Normality is not transitive
 * Normality satisfies image condition
 * Normality satisfies inverse image condition

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 $$K$$.