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, ‎$mathcal{S}$ 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}.]‎

Abstract licence: CC BY

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