$p$-elements in profinite groups
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913407285854208 |
|---|---|
| author | Lucchini, Andrea Otmen, Nowras |
| author_facet | Lucchini, Andrea Otmen, Nowras |
| contents | We investigate some properties of the $p$-elements of a profinite group $G$. We prove that if $p$ is odd and the probability that a randomly chosen element of $G$ is a $p$-element is positive, then $G$ contains an open prosolvable subgroup. On the contrary, there exist groups that are not virtually prosolvable but in which the probability that a randomly chosen element of $G$ is a 2-element is arbitrarily close to 1. We prove also that if a profinite group $G$ has the property that, for every $p$-element $x$, it is positive the probability that a randomly chosen element $y$ of $G$ generates with $x$ a pro-$p$ group, then $G$ contains an open pro-$p$ subgroup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_19766 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $p$-elements in profinite groups Lucchini, Andrea Otmen, Nowras Group Theory We investigate some properties of the $p$-elements of a profinite group $G$. We prove that if $p$ is odd and the probability that a randomly chosen element of $G$ is a $p$-element is positive, then $G$ contains an open prosolvable subgroup. On the contrary, there exist groups that are not virtually prosolvable but in which the probability that a randomly chosen element of $G$ is a 2-element is arbitrarily close to 1. We prove also that if a profinite group $G$ has the property that, for every $p$-element $x$, it is positive the probability that a randomly chosen element $y$ of $G$ generates with $x$ a pro-$p$ group, then $G$ contains an open pro-$p$ subgroup. |
| title | $p$-elements in profinite groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2406.19766 |