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.



Does $\chi(1)^2=|G:Z(G)|$ for irreducible character of a finite group $G$ imply $G$ is solvable?
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**](http://dx.doi.org/10.1007/BF01215066), Math. Z. 179, 555-569 (1982). [MR652860](https://mathscinet.ams.org/mathscinet-getitem?mr=652860) [ZBL0511.20002](https://zbmath.org/?q=an:0511.20002).


The proof uses the classification of finite simple groups.


