Let $G$ be a group and let $N \le G$ be a subgroup. For $g \in G$, write $gN := \{gn : n \in N\}$, $Ng := \{ng : n \in N\}$, and $gNg^{-1} := \{gng^{-1} : n \in N\}$. The following conditions are equivalent:
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1. $N \trianglelefteq G$.
2. For every $g \in G$, one has $gNg^{-1} \subset N$.
3. For every $g \in G$, one has $gN = Ng$.
4. For every $g \in G$ and every $n \in N$, one has $gng^{-1} \in N$.