TITLE: character degree and solvability
QUESTION [6 upvotes]: There is an unsolved problem in Berkovich's book "Characters of Finite Groups Part 2" I state here:
Is $G$ solvable if $\chi(1)^2$ divides $|G|$ for all $\chi \in {\rm Irr}(G)$?
Can any one tell me some latest progresses for this? Maybe you can tell me some latest research papers. Thank you.

REPLY [5 votes]: Steve Gagola and I have a paper where we address this very question.  I am away from home and so I don't have the reference handy.  It was in the Communications in Algebra in the late 90's.  I think 1998.
Also, the example Marty gave was in that paper.  Marty and I found that example while I was still working on my PhD.