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> [!theorem] Theorem. ([[conjugate characterization of normal subgroups]])
> Let $G$ be a [[group]] and $N$ [[subgroup]]. The following are equivalent:
> - $N$ is [[normal subgroup|normal]] (i.e., $gN=Ng$ for all $g \in G$);
> - $gNg^{-1}=N$ for all $g \in G$;
> - $g^{-1} N g = N$ for all $g \in G$;
> - $g N g^{-1} \subset N$ for all $g \in G$;
> - $N \subset g N g^{-1}$ for all $g \in G$;
> - $gN \subset Ng$ for all $g \in G$;
> - Every [[coset|left coset]] of $N$ in $G$ is a [[coset|right coset]];
> - Every [[coset|right coset]] of $N$ in $G$ is a [[coset|left coset]].
>
^53879a
> [!proof]- Proof. ([[conjugate characterization of normal subgroups]])
> It is clear that $(1) \implies (2) \implies (3) \implies (4) \implies (5) \implies (6)$. 'Every left coset is a right coset' means that for all $a \in G$, $aN=Nb$ for some $b \in G$. Since $a=ae \in aN$, certainly $a \in Nb$. But since [[Lagrange's Theorem|right cosets partition]] $G$ and $a \in Na$, the only right coset containing $a$ is $Na$... so $a=b$.
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#### References
> [!backlink]
> ```dataview
TABLE rows.file.link as "Further Reading"
FROM [[]]
FLATTEN file.tags
GROUP BY file.tags as Tag
> [!frontlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM outgoing([[]])
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```