$p$-groups with small number of character degrees and their normal subgroups
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911809334673408 |
|---|---|
| author | Talukdar, Nabajit Rajkhowa, Kukil Kalpa |
| author_facet | Talukdar, Nabajit Rajkhowa, Kukil Kalpa |
| contents | If $G$ be a finite $p$-group and $χ$ is a non-linear irreducible character of $G$, then $χ(1)\leq |G/Z(G)|^{\frac{1}{2}}$. In \cite{fernandez2001groups}, Fernández-Alcober and Moretó 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 $cd(G) \subseteq \{1, p^{n-1}, p^{n}\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_15094 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $p$-groups with small number of character degrees and their normal subgroups Talukdar, Nabajit Rajkhowa, Kukil Kalpa Group Theory 20C15 If $G$ be a finite $p$-group and $χ$ is a non-linear irreducible character of $G$, then $χ(1)\leq |G/Z(G)|^{\frac{1}{2}}$. In \cite{fernandez2001groups}, Fernández-Alcober and Moretó 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 $cd(G) \subseteq \{1, p^{n-1}, p^{n}\}$. |
| title | $p$-groups with small number of character degrees and their normal subgroups |
| topic | Group Theory 20C15 |
| url | https://arxiv.org/abs/2403.15094 |