**Seminario Teoría de Grupos UAM-ICMAT **

**Fecha: Jueves 11 de Abril, 2019 **

**Hora: 11:30 **

**Lugar: Aula 520 UAM **

**Speaker: Joan Tent (Universidad de Valencia) **

**Title: Finite groups with character values in $mathbb Q_p$ **

**Abstract: A classical problem in character theory of finite groups consists in showing how rationality properties **

**of characters and conjugacy classes of finite groups are reflected in the structure of a group. **

**A well-known theorem by R. Gow in this setting establishes that if $G$ is a finite rational solvable group and $ell$ is a prime divisor of the order of $G$, **

**then $ellin{2, 3, 5}$, thus determining the possible composition factors of $G$. **

**Our aim in this talk is to present an odd analogue of Gow's theorem: if all characters of a solvable group $G$ take values **

**in the field $mathbb Q_p=mathbb Q(xi)$, where $xiinmathbb C^times$ has prime order $o(xi)=p>2$, then **

**the prime divisors of the order of $G$ lie in the set ${2,3,5, p, 2p+1}$. **

**We shall also discuss possible generalizations to non-solvable finite groups. **