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For questions concerning normal subgroups of groups. Consider using with the (group-theory) and/or the (abstract-algebra) tags too.

A subgroup $N$ of a group $G$ is called normal if the sets of left and right cosets of the subgroup coincide. This can be equivalently characterized a few different ways. A subgroup $N$ of $G$ is normal in $G$ if either of the following are true:

  • For any $g,h \in G$, if $gh \in N$ then $hg \in N$.
  • We have that $gN=Ng$ for all $g\in G$.
  • For each $n\in N$ and each $g\in G$, we have $gng^{-1}\in N$.
  • $G/N$, the collection of left cosets of $N$ in $G$, inherits a well-defined group structure from the operation on $G$. This entails the left and right cosets coinciding, so we can dually require that the set of right cosets $N\!\setminus\!G$ inherits a well-defined group structure from the operation on $G$.
  • $N$ is the kernel of a group homomorphisms $\phi\colon G \to H$ for some other group $H$. Furthermore, in view of the first isomorphism theorem, the image of $G$ in $H$ will be isomorphic to the quotient $G/N$.

These last two bullets suggest the intuition that normal subgroups are exactly the subgroups that you may quotient by. When $N$ is a normal subgroup of $G$, this is often denoted by writing $N \vartriangleleft G$.