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Locally graded groups with a condition on infinite subsets | ||
| International Journal of Group Theory | ||
| مقاله 1، دوره 7، شماره 4، اسفند 2018، صفحه 1-7 اصل مقاله (203.76 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2016.21234 | ||
| نویسندگان | ||
| Asadollah Faramarzi Salles* ؛ Fatemeh Pazandeh Shanbehbazari | ||
| Damghan University | ||
| چکیده | ||
| 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 $x\neq y, z$ 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 \[G\in\mathcal{L}\Leftrightarrow G\in\mathcal{R}\Leftrightarrow G\in\mathcal{S}\Leftrightarrow G\in\mathcal{C}\Leftrightarrow G\in\mathcal{N}_2\mathcal{F}.\] | ||
| کلیدواژهها | ||
| Finitely generated groups؛ Residually finite groups؛ Locally graded groups | ||
| مراجع | ||
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