Element orders in extraspecial groups
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913343358369792 |
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| author | Lazorec, Mihai-Silviu |
| author_facet | Lazorec, Mihai-Silviu |
| contents | By using the structure and some properties of extraspecial and generalized/almost extraspecial $p$-groups, we explicitly determine the number of elements of specific orders in such groups. As a consequence, one may find the number of cyclic subgroups of any (generalized/almost) extraspecial group. For a finite group $G$, the ratio of the number of cyclic subgroups to the number of subgroups is called the cyclicity degree of $G$ and is denoted by $cdeg(G)$. We show that the set containing the cyclicity degrees of all finite groups is dense in $[0, 1]$. This is equivalent to giving an affirmative answer to the following question posed by Tóth and Tărnăuceanu: ``For every $a\in [0, 1]$, does there exist a sequence $(G_n)_{n\geq 1}$ of finite groups such that $\displaystyle\lim_{n\to\infty} cdeg(G_n)=a$?". We show that such sequences are formed of finite direct products of extraspecial groups of a specific type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_04141 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Element orders in extraspecial groups Lazorec, Mihai-Silviu Group Theory By using the structure and some properties of extraspecial and generalized/almost extraspecial $p$-groups, we explicitly determine the number of elements of specific orders in such groups. As a consequence, one may find the number of cyclic subgroups of any (generalized/almost) extraspecial group. For a finite group $G$, the ratio of the number of cyclic subgroups to the number of subgroups is called the cyclicity degree of $G$ and is denoted by $cdeg(G)$. We show that the set containing the cyclicity degrees of all finite groups is dense in $[0, 1]$. This is equivalent to giving an affirmative answer to the following question posed by Tóth and Tărnăuceanu: ``For every $a\in [0, 1]$, does there exist a sequence $(G_n)_{n\geq 1}$ of finite groups such that $\displaystyle\lim_{n\to\infty} cdeg(G_n)=a$?". We show that such sequences are formed of finite direct products of extraspecial groups of a specific type. |
| title | Element orders in extraspecial groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2405.04141 |