Abstract Nonsense

Crushing one theorem at a time

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.

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March 7, 2011 Posted by | Algebra, Group Theory, Representation Theory | , , , , , | 4 Comments