| Abstract |
Representation Theory of finite groups through matrices over complex num- bers was initiated primarily by G. Frobenius, with the significant contribution of I. Schur. A great amount of Frobenius’ results were proven independently from W. Burnside with the use of characters of groups. Through E. Noe- ther’s contribution, representation theory was developed following a different direction using representation modules. Today, Representation Theory, and especially Character Theory, is one of the most fundamental fields of Mathe- matics, useful for the easy solution of a wide range of problems in various mathematical sectors besides Group Theory, two typical examples of such problems being Burnside’s paqb theorem and the form of Frobenius groups.
According to Burnside’s paqb theorem, each group of order paqb is solva- ble (where p, q are prime numbers). The theory of the characters of a finite group was utilized for this theorem’s proof. The same applies for the chara- cterization of Frobenius groups. However, even though when it comes to the Burnside theorem, there exists a proof without the use of characters (which is significantly more difficult), the same is not true for Frobenius groups. E- ven until today, the sole proof we have for the characterization of Frobenius groups is based on the characters of these groups (as we will examine in the second chapter of this thesis).
The irreducible characters of a group possess many noteworthy attributes and comprise crucial calculation tools in Representation Theory. By studying only these characters, and not the representations they originate from, one can deduct results about the constitution of the group itself. In this thesis, we will focus on the zeros of non-linear characters and the results that arise for the characters as well as for the group, if we are aware of the amount of the zeros of its reducible characters or their mean value.
One of the first results relative to the zeros of characters was presented by Burnside himself, confirming that they exist:
“If G is a finite group and χ is a non-linear irreducible character of G, then there exists g ∈ G such that χ(g) = 0.”
In other words, every (non-linear) irreducible character of a group vani- shes in at least one element of the group and thus in its entire conjugacy class. The way in which the amount of the zeros of the irreducible characters of a group affects its composition has been studied by Chillag [1], Madanha [9], Navarro [10] and Yang [12], whose most significant results were examined and assembled in this thesis. Therefore, for example, we prove:
Let N :9 G and χ ∈ Irr(G). Then χN is not irreducible if and only if χ
vanishes on a coset N x of N in G.
If anz(G)< 1, then G is solvable (where we write anz(G) for the quotient of the number of zeros on the character table of G and the number of its irreducible characters).
If anz(G) < 1 , then G is supersolvable.
The first chapter is dedicated to Representation Theory and its most important theorems. We display the connection between the representations of a group G and F G-modules, linear spaces over a field F on which we define an operation between the elements of the module and the elements of G. Furthermore, we give a definition for the character of a group and exhibit its basic properties. The chapter is concluded with the fundamental theorems of Clifford Theory, which involve characters induced from a normal subgroup of our group. Most results from this chapter can be found in [8]. For further information on Clifford Theory, we also suggest [4].
In the second chapter, we study two specific subjects of the theory in de- pth: the Frobenius groups and the character triples. Specifically, we define when G is a Frobenius group, we prove the existence of the Frobenius Kernel and we calculate all of the group’s irreducible characters. Furthermore, we thoroughly examine character triples, we define when two character triples are isomorphic and we prove that for each character triple there exists an isomorphic one with “good” properties. Finally, using character triples, we present a sufficient condition which allows us to expand a irreducible chara- cter of a normal subgroup to an irreducible character of the group itself. For the contents of this chapter we refer to [4], [7] and [12].
The theorems that concern the number of zeros of the characters of a group are exhibited in the third and final chapter. Using the articles mentio- ned above, we present and prove their most significant results.
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