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On a graph associated to groups. (English)
Bull. Malays. Math. Sci. Soc. (2) 34, No. 3, 553-560 (2011).
Summary: Let $G$ be a group such that the subgroup $T^2(G)=\{x\in G\mid (gx)^2=(xg)^2$, for all $g\in G\}$ is a proper subgroup. We define a graph $Γ$ with the vertex set $G\setminus T^2(G)$ in which two vertices $x,y$ are joined by an edge if $(xy)^2=(yx)^2$. In this note we study this graph and find a characterization of some dihedral groups in terms of this graph.