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A generalization of Taketa's theorem on $\rm M$-groups II | ||
| AUT Journal of Mathematics and Computing | ||
| مقاله 7، دوره 4، شماره 1، 2023، صفحه 63-67 اصل مقاله (337.67 K) | ||
| نوع مقاله: Original Article | ||
| شناسه دیجیتال (DOI): 10.22060/ajmc.2022.21781.1108 | ||
| نویسنده | ||
| Zeinab Akhlaghi* | ||
| Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran | ||
| چکیده | ||
| 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 $\mathrm{acd}_{nm}(G)< \mathrm{acd}_{nm}(\mathrm{SL}_2(5))=19/7$. | ||
| کلیدواژه‌ها | ||
| Monomial character؛ Primitive character؛ Taketa’ s theorem؛ Average degree | ||
| مراجع | ||
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[1] Z. Akhlaghi, A generalization of taketa’s theorem on M-groups, Quaestiones Mathematicae, (2022), pp. 1–6.
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[5] I. M. Isaacs, Character theory of finite groups, Dover Publications, Inc., New York, 1994. Corrected reprint of the 1976 original [Academic Press, New York].
[6] T. Le, J. Moori, and H. P. Tong-Viet, On a generalization of M-group, J. Algebra, 374 (2013), pp. 27–41.
[7] A. Moreto and H. N. Nguyen ´ , On the average character degree of finite groups, Bull. Lond. Math. Soc., 46 (2014), pp. 454–462.
[8] G. Qian, Two results related to the solvability of M-groups, J. Algebra, 323 (2010), pp. 3134–3141.
[9] , On the average character degree and the average class size in finite groups, J. Algebra, 423 (2015), pp. 1191–1212.
[10] , Nonsolvable groups with few primitive character degrees, J. Group Theory, 21 (2018), pp. 295–318. | ||
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