## Thesis etd-12062011-191704 |

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Thesis type

Tesi di dottorato di ricerca

Author

MARINELLI, SELENA

URN

etd-12062011-191704

Thesis title

Zeros and degrees of characters of
finite groups

Academic discipline

MAT/02

Course of study

MATEMATICA

Supervisors

**tutor**Prof. Dolfi, Silvio

**relatore**Prof. Navarro, Gabriel

Keywords

- Character theory
- irreducible character degrees
- real irreducible characters
- Vanishing and non-vanishing elements

Graduation session start date

16/12/2011

Availability

Full

Summary

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.

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.

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