TITLE: Does $\chi(1)^2=|G:Z(G)|$ for irreducible character of a finite group $G$ imply $G$ is solvable?
QUESTION [11 upvotes]: In "Character Theory of Finite Groups" I.M. Isaacs mention the following conjecture:
It is only possible in a solvable group $G$ to have $\chi(1)^2=|G:Z(G)|$ with $\chi \in$ Irr$(G)$.
Is this problem still open? I tried to search for attempts to solve it but didn't find anything.

REPLY [13 votes]: Such groups are solvable. This has been solved by Howlett and Isaacs himself, in
Howlett, Robert B.; Isaacs, I. Martin, On groups of central type, Math. Z. 179, 555-569 (1982). MR652860 ZBL0511.20002.
The proof uses the classification of finite simple groups.