Normal Closure

Let $G$ be a group and let $A\subseteq G$. The normal closure of $A$ in $G$, denoted $\langle\!\langle A\rangle\!\rangle$, is the smallest normal subgroup of $G$ containing $A$. Equivalently,

the intersection of all normal subgroups containing $A$.

A concrete description is: $\langle\!\langle A\rangle\!\rangle$ is the subgroup generated by all conjugates $gag^{-1}$ with $g\in G$ and $a\in A$ (so it is the smallest normal subgroup closed under conjugation containing $A$), which ties it to the conjugation action

Examples:

• In $S_3$, the normal closure of a transposition is all of $S_3$ (because conjugates of a transposition are all transpositions, and these generate $S_3$).
• In $S_3$, the normal closure of a $3$-cycle is $A_3$.
• If $G$ is abelian, then every subgroup is normal, so the normal closure of $A$ is just the subgroup generated by $A$.