%0 Journal Article
%T A generalization of Taketa's theorem on $\rm M$-groups II
%J AUT Journal of Mathematics and Computing
%I Amirkabir University of Technology
%Z 2783-2449
%A Akhlaghi, Zeinab
%D 2023
%\ 02/01/2023
%V 4
%N 1
%P 63-67
%! A generalization of Taketa's theorem on $\rm M$-groups II
%K Monomial character
%K Primitive character
%K Taketaâ€™ s Theorem
%K Average degree
%R 10.22060/ajmc.2022.21781.1108
%X 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 $\mathrm{acd}_{nm}(G)$, 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 $\mathrm{acd}_{nm}(G)< \mathrm{acd}_{nm}(\mathrm{SL}_2(5))=19/7$.
%U https://ajmc.aut.ac.ir/article_5011_0057069c3de974695209b73b6eb0947d.pdf