It is known that, if all the irreducible real valued characters of a finite
group are of odd degree, then the group has a normal Sylow 2-subgroup. In this
paper, we prove and analogous result for solvable groups, by taking into
account the degree of irreducible characters fixed by some field isomorphism of
prime order $p$.
  We prove it as a consequence of a more general result

Grittini, Nicola

English

arXiv

It is known that, if all the real-valued irreducible characters of a finite
group have odd degree, then the group has normal Sylow $2$-subgroup. We
generalize this result for Sylow $p$-subgroups, for any prime number $p$, while
assuming the group to be $p$-solvable. In particular, it is proved that a
$p$-solvable group has a normal Sylow $p$-subgroup if $p$ does not divide the
degree of any irreducible character of the group fixed by a field automorphism
of order $p$

On the degrees of irreducible characters fixed by some field automorphism

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