# Tesi etd-12062011-191704

Thesis type
Tesi di dottorato di ricerca
Author
MARINELLI, SELENA
URN
etd-12062011-191704
Title
Zeros and degrees of characters of finite groups
Settore scientifico disciplinare
MAT/02
Corso di studi
MATEMATICA
Commissione
tutor Prof. Dolfi, Silvio
relatore Prof. Navarro, Gabriel
Parole chiave
• Vanishing and non-vanishing elements
• real irreducible characters
• irreducible character degrees
• Character theory
Data inizio appello
16/12/2011;
Consultabilità
completa
Riassunto analitico
This thesis addresses some questions about the relationship between the structure of finite groups and the set of their character degrees (Chapters 2 and 5) and the set of their vanishing elements (Chapters 3 and 4).<br><br>In Chapter 2,we study the groups G such that <br>whenever $1\neq\chi,\psi \in$ Irr(G) have the same degree,then there exists an automorphism $\alpha$ of G such that $\chi=\psi^{\alpha}$. We call AC the class of such groups G. In Section 2.2, we give a tight description of the <br>structure of solvable groups in AC. In Section 2.3, we discuss the problem of determining all the simple groups in AC. <br><br>In Chapter 3, we generalize some recent results regarding the relationship between the structure of a group and his vanishing elements.<br><br>In Chapter 4, we solve a new problem concerning vanishing elements and characters with prescribed fields of values. Let F be a subfield of $\mathbb{C}$, we write Irr$_{F}$(G) for the set of those $\chi$ in Irr(G) such that $\chi(g) \in F$ for all $g \in G$. Then we consider $F=\mathbb{R}$ and the set of real-valued (or real) irreducible characters of G.<br>The main goal of Chapter 4 states that if $\chi(x) \neq 0$ for all real $\chi$ in Irr(G) and all 2-elements x in G, then<br>G has a normal Sylow 2-subgroup.<br><br>Finally, in Chapter 5, we come back to an old problem by R. Gow (in 1975), on the groups whose irreducible character degrees can be linearly ordered by divisibility. Gow&#39;s main result is not complete: there are three undecided cases. In Chapter 5 we will settle two of these three cases.
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