Inner Automorphism Maps Subgroup to Itself iff Normal/Necessary Condition

Theorem

Let $G$ be a group.

For $x \in G$, let $\kappa_x$ denote the inner automorphism of $x$ in $G$.

Suppose that:

$\forall x \in G: \kappa_x \sqbrk H = H$

Then $H$ is a normal subgroup of $G$.

Proof

Suppose that:

$\forall x \in G: \kappa_x \sqbrk H = H$

Let $x \in G$ be arbitrary.

By definition of inner automorphism of $x$ in $G$:

$\forall h \in H: x h x^{-1} \in H$

So, by definition, $H$ is a normal subgroup of $G$

$\blacksquare$