Locally graded groups with a condition on infinite subsets
| Autor: | Asadollah Faramarzi Salles, Fatemeh Pazandeh Shanbehbazari |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: | |
| Zdroj: | International Journal of Group Theory, Vol 7, Iss 4, Pp 1-7 (2018) |
| Druh dokumentu: | article |
| ISSN: | 2251-7650 2251-7669 |
| DOI: | 10.22108/ijgt.2016.21234 |
| Popis: | Let $G$ be a group, we say that $G$ satisfies the property $mathcal{T}(infty)$ provided that, every infinite set of elements of $G$ contains elements $xneq y, z$ such that $[x, y, z]=1=[y, z, x]=[z, x, y]$. We denote by $mathcal{C}$ the class of all polycyclic groups, $mathcal{S}$ the class of all soluble groups, $mathcal{R}$ the class of all residually finite groups, $mathcal{L}$ the class of all locally graded groups, $mathcal{N}_2$ the class of all nilpotent group of class at most two, and $mathcal{F}$ the class of all finite groups. In this paper, first we shall prove that if $G$ is a finitely generated locally graded group, then $G$ satisfies $mathcal{T}(infty)$ if and only if $G/Z_2(G)$ is finite, and then we shall conclude that if $G$ is a finitely generated group in $mathcal{T}(infty)$, then [Ginmathcal{L}Leftrightarrow Ginmathcal{R}Leftrightarrow Ginmathcal{S}Leftrightarrow Ginmathcal{C}Leftrightarrow Ginmathcal{N}_2mathcal{F}.] |
| Databáze: | Directory of Open Access Journals |
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