Normality is quotient-transitive

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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

• 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

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 $K$.