Locally graded groups with a condition on infinite subsets
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{...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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University of Isfahan
2018-12-01
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| Series: | International Journal of Group Theory |
| Subjects: | |
| Online Access: | http://ijgt.ui.ac.ir/article_21234_67c122bc31064ada379ba0fa8178aec3.pdf |
| Summary: | 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}.] |
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| ISSN: | 2251-7650 2251-7669 |