Generator of Normal Subgroup

Theorem
Let $G$ be a group.

Let $S \subseteq G$.

Let $\hat S = S \cup S^{-1}$, where $S^{-1}$ is the set of all the inverses of all the elements of $S$.

Let $\tilde S = \left\{{a s a^{-1}: s \in \hat S, a \in G}\right\}$.

Let $W \left({\tilde S}\right)$ be the set of words of $\tilde S$.

Let $N$ be the smallest normal subgroup of $G$ that contains $S$.

Then $N = \left \langle {S} \right \rangle = W \left({\tilde S}\right)$.

Proof
Let $N$ be the smallest normal subgroup of $G$ that contains $S$, where $S \subseteq G$.


 * $N$ must certainly include $\hat S$, because any group containing $s \in S$ must also contain $s^{-1}$.

Therefore, $N$ must be the smallest normal subgroup containing $\hat S$.

Since $N \triangleleft G$, it follows that $\forall a \in G: \forall s \in \hat S: a s a^{-1} \in N$.

Thus, $\forall x \in \tilde S: x \in N$.

Thus $\tilde S \subseteq N$.


 * By the closure axiom, $N$ must also contain all products of any finite number of elements of $\tilde S$. Thus $W \left({\tilde S}\right) \subseteq N$.


 * Now we prove that $W \left({\tilde S}\right) \triangleleft G$.

$\tilde S \ne \varnothing$, as $e \left({s s^{-1}}\right) e^{-1} = e \in \tilde S$.

By Conjugate of Set with Inverse Closed for Inverses, $\tilde S$ is closed under taking inverses.

So from Set of Words Generates Group: Corollary, $W \left({\tilde S}\right) \le G$.

From Conjugate of Set with Inverse is Closed, $\forall w \in W \left({\tilde S}\right): \forall a \in G: a w a^{-1} \in W \left({\tilde S}\right)$.


 * From Normal Subgroup Equivalent Definitions: 3, it follows that $W \left({\tilde S}\right) \triangleleft G$.

The result follows by the definition of the minimality of $N$.