Normal Closure - Knowlpedia

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

\langle\!\langle A\rangle\!\rangle \;=\; \bigcap\{N\trianglelefteq G : A\subseteq N\},

the intersection of all normal subgroups containing $A$ .

It is also the subgroup generated by the conjugates $gag^{-1}$ with $g\in G$ and $a\in A$ .

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 $A$ .