Zeros and degrees of characters of finite groups

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). In Chapter 2,we study the groups G such that 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 structure of solvable groups in AC. In Section 2.3, we discuss the problem of determining all the simple groups in AC. In Chapter 3, we generalize some recent results regarding the relationship between the structure of a group and his vanishing elements. 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. 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 G has a normal Sylow 2-subgroup. 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's main result is not complete: there are three undecided cases. In Chapter 5 we will settle two of these three cases.