Tag Info

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 $$N \vartriangleleft G$$.