On a question of Jaikin-Zapirain about the average order elements of finite groups
| Autor: | Bijan Taeri, Ziba Tooshmalani |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2024 |
| Předmět: | |
| Zdroj: | International Journal of Group Theory, Vol 14, Iss 3, Pp 139-147 (2024) |
| Druh dokumentu: | article |
| ISSN: | 2251-7650 2251-7669 |
| DOI: | 10.22108/ijgt.2024.139508.1879 |
| Popis: | For a finite group $G$, the average order $o(G)$ is defined to be the average of all order elements in $G$, that is $o( G)=\frac{1}{|G|}\sum_{x\in G}o(x)$, where $o(x)$ is the order of element $x$ in $G$. Jaikin-Zapirain in [On the number of conjugacy classes of finite nilpotent groups, Advances in Mathematics, \textbf{227} (2011) 1129-1143] asked the following question: if $G$ is a finite ($p$-) group and $N$ is a normal (abelian) subgroup of $G$, is it true that $o(N)^{\frac{1}{2}}\leq o(G) $? We say that $G$ satisfies the average condition if $o(H)\leq o(G)$, for all subgroups $H$ of $G$. In this paer we show that every finite abelian group satisfies the average condition. This result confirms and improves the question of Jaikin-Zapirain for finite abelian groups. |
| Databáze: | Directory of Open Access Journals |
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