A generalization of Taketa's theorem on $\rm M$-groups $\rm II$

Document Type : Original Article


Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran


In the recent paper [A generalization of Taketa's theorem on $M$-groups, Quaestiones Mathematicae, (2022)], we give an upper bound $5/2$ for the average of non-monomial character degrees of a finite group $G$, denoted by ${\rm acd}_{nm}(G)$, which guarantees the solvability of $G$. Although the result is true, the example we gave to show that the bound is sharp turns out to be incorrect. In this paper we find a new bound and we give an example to show that this new bound is sharp. Indeed, we prove the solvability of $G$, by assuming ${\rm acd}_{nm}(G)<{\rm acd}_{nm}(SL_2(5))=19/7$.


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