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$p$-groups with a small number of character degrees and their normal subgroups | ||
| International Journal of Group Theory | ||
| مقاله 7، دوره 14، شماره 3، آذر 2025، صفحه 171-180 اصل مقاله (431.13 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2024.141029.1897 | ||
| نویسندگان | ||
| Nabajit Talukdar* ؛ Kukil Kalpa Rajkhowa | ||
| Department of Mathematics, Cotton University, Guwahati, India | ||
| چکیده | ||
| If $G$ be a finite $p$-group and $\chi$ is a non-linear irreducible character of $G$, then $\chi(1)\leq |G/Z(G)|^{\frac{1}{2}}$. In \cite{fernandez2001groups}, Fern'{a}ndez-Alcober and Moret'{o} obtained the relation between the character degree set of a finite $p$-group $G$ and its normal subgroups depending on whether $|G/Z(G)|$ is a square or not. In this paper we investigate the finite $p$-group $G$ where for any normal subgroup $N$ of $G$ with $G'\not \leq N$ either $N\leq Z(G)$ or $|NZ(G)/Z(G)|\leq p$ and obtain some alternate characterizations of such groups. We find that if $G$ is a finite $p$-group with $|G/Z(G)|=p^{2n+1}$ and $G$ satisfies the condition that for any normal subgroup $N$ of $G$ either $G'\not \leq N$ or $N\leq Z(G)$, then $cd(G)=\{1, p^{n}\}$. We also find that if $G$ is a finite $p$-group with nilpotency class not equal to $3$ and $|G/Z(G)|=p^{2n}$ and $G$ satisfies the condition that for any normal subgroup $N$ of $G$ either $G'\not \leq N$ or $|NZ(G)/Z(G)|\leq p$, then $cd(G) \subseteq \{1, p^{n-1}, p^{n}\}$. | ||
| کلیدواژهها | ||
| Character degrees؛ $p$-groups؛ nilpotency class | ||
| مراجع | ||
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[1] H. Doostie and A. Saeidi, Finite p-groups with few non-linear irreducible character kernels, Bull. Iranian Math. Soc., 38 no. 2 (2012) 413–422. | ||
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آمار تعداد مشاهده مقاله: 516 تعداد دریافت فایل اصل مقاله: 562 |
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