On the number of conjugacy classes of subgroups of a finite group
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866908495473803264 |
|---|---|
| author | Lazorec, Mihai-Silviu Tărnăuceanu, Marius |
| author_facet | Lazorec, Mihai-Silviu Tărnăuceanu, Marius |
| contents | Let $k'(G)$ and $L(G)$ be the number of conjugacy classes of subgroups and the subgroup lattice of a finite group $G$, respectively. Our objective is to study some aspects related to the ratios $d'(G)=\frac{k'(G)}{|L(G)|}$ and $d^*(G)=\min\{ d'(S) \mid S\text{ is a section of }G\}$ which measure how close is $G$ from being a Dedekind group. We prove that the set containing the values $d'(G)$, as $G$ ranges over the class of nilpotent groups, is dense in $[0, 1]$. A nilpotency criterion is obtained by proving that if $d^*(G)>\frac{2}{3}$, then $G$ is nilpotent and information on its structure is given. We also show that if $d^*(G)>\frac{4}{5}$, then $G$ is an Iwasawa group. Finally, we deduce a result which ensures that a $p$-group of order $p^n$ ($n\geq 3$) is a Dedekind group. This last result can be extended to the class of nilpotent groups and it also highlights the second maximum values of $d'$ and $d^*$ on the class of $p$-groups of order $p^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_14267 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the number of conjugacy classes of subgroups of a finite group Lazorec, Mihai-Silviu Tărnăuceanu, Marius Group Theory Let $k'(G)$ and $L(G)$ be the number of conjugacy classes of subgroups and the subgroup lattice of a finite group $G$, respectively. Our objective is to study some aspects related to the ratios $d'(G)=\frac{k'(G)}{|L(G)|}$ and $d^*(G)=\min\{ d'(S) \mid S\text{ is a section of }G\}$ which measure how close is $G$ from being a Dedekind group. We prove that the set containing the values $d'(G)$, as $G$ ranges over the class of nilpotent groups, is dense in $[0, 1]$. A nilpotency criterion is obtained by proving that if $d^*(G)>\frac{2}{3}$, then $G$ is nilpotent and information on its structure is given. We also show that if $d^*(G)>\frac{4}{5}$, then $G$ is an Iwasawa group. Finally, we deduce a result which ensures that a $p$-group of order $p^n$ ($n\geq 3$) is a Dedekind group. This last result can be extended to the class of nilpotent groups and it also highlights the second maximum values of $d'$ and $d^*$ on the class of $p$-groups of order $p^n$. |
| title | On the number of conjugacy classes of subgroups of a finite group |
| topic | Group Theory |
| url | https://arxiv.org/abs/2508.14267 |