## 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 and the set of all normal subgroups of . In particular, we prove that every normal subgroup of has the realization as the kernel of some character, and thus by previous theorem that every normal subgroup of has a realization as the intersection of certain ‘s.

*Motivation*

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