Inner Automorphism Maps Subgroup to Itself iff Normal/Necessary Condition
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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 $G$
$\blacksquare$