Normal Subgroup Test

Theorem
Let $$G$$ be a group and $$H \le G$$.

Then $$H$$ is a normal subgroup of $$G$$ iff:

$$\forall x \in G: x H x^{-1} \subseteq H$$.

Proof
Let $$H$$ be a subgroup of $$G$$.


 * Suppose $$H$$ is normal in $$G$$.

Then $$\forall x \in G, a \in H: \exists b \in H: x a = b x$$.

Thus, $$x a x^{-1} = b \in H$$ implying $$x H x^{-1} \subseteq H$$.


 * Conversely, suppose $$\forall x \in G: x H x^{-1} \subseteq H$$.

Then for $$g \in G$$, we have $$g H g^{-1} \subseteq H$$, which implies $$g H \subseteq H g$$.

Also, for $$g^{-1} \in G$$, we have $$g^{-1} H (g^{-1})^{-1} = g^{-1} H g \subseteq H$$ which implies $$H g \subseteq g H$$.

Therefore, $$g H = H g$$ meaning $$H \triangleleft G$$.