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edited Mar 18, 2020 at 3:37

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gr.group-theory finite-groups solvable-groups character-theory

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asked Mar 18, 2020 at 0:55

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Does $\chi(1)^2=|G:Z(G)|$ for irreducible character of a finite group $G$ imply $G$ is solvable?

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.

finite-groups solvable-groups character-theory