Sorry for any mistakes I make here, this is my first post here. I have a group $G$ which has an abelian subgroup $A<G$. I also know there is a irreducible character $\chi$ with the degree of $\chi$ equal to the index of $A$ in $G$. This implies $G$ has a non-identity abelian NORMAL subgroup? How?
This is exercise 2.17 from Isaacs's Character Theory of Finite Groups, page 31.
The hint is to show that $\chi$ vanishes on $G-A$.