Normal subgroup/Characterization/Fact/Proof

Proof

(1) means for given ${}h\in H$ that we can write ${}xh={\tilde {h}}x$ with some ${}{\tilde {h}}\in H$. Multiplication by ${}x^{-1}$ from the right yields ${}xhx^{-1}={\tilde {h}}\in H$; therefore, ${}(2)$ holds. Reading this argument backwards gives the implication ${}(2)\Rightarrow (1)$. Moreover, ${}(2)$ is an explicit reformulation of ${}(3)$.

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