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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.