TITLE: character degree and solvability
QUESTION [7 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 $\chi \in \operatorname{Irr}{(G)}$?

Can any one tell me some latest progresses for this? Maybe you can tell me some latest research papers. Thank you.

REPLY [2 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. Mark Lewis
Also, the example Marty gave was in that paper. Marty and I found that example while I was still working on my PhD.