# Abstract Nonsense

## Relation Between the Kernels of Characters and Normal Subgroups

Point of post: In this post we derive some fundamental relationships between the kernels of the characters of a finite group $G$ and the set of all normal subgroups of $G$. In particular, we prove that every normal subgroup of $G$ has the realization as the kernel of some character, and thus by previous theorem that every normal subgroup of $G$ has a realization as the intersection of certain $N^{(\alpha)}$‘s.

Motivation

We saw in our last post that there is a particularly fruitful way to produce normal subgroups ofa finite group $G$. Namely, one can look at the kernels of the characters of $G$, which as was shown amounts (in the sense that all the necessary information is contained in) to finding the kernels of the irreducible characters of $G$. What we shall now show is that in fact, finding the kernels of the irreducible characters of a group $G$ enables us to list off precisely what normal subgroups $G$ has. In particular, we will see that every normal subgroup of $G$ has a realization as the kernel of a certain character. We shall do this by constructing a certain character on the quotient group of $G$ for which, when thought of a character of $G$, will have kernel equal to the normal subgroup.