Normal Subgroup Test
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Theorem
Let $G$ be a group and $H \le G$.
Then $H$ is a normal subgroup of $G$ if and only if:
- $\forall x \in G: x H x^{-1} \subseteq H$.
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 $H \lhd G$.
$\blacksquare$
