Journal of the
Korean Mathematical Society JKMS

Let $G\otimes G$ be the tensor square of a group $G$. The set of all elements $a$ in $G$ such that $a\otimes g=1_{\otimes}$, for all $g$ in $G$, is called the tensor centre of $G$ and denoted by $Z^{\otimes}(G)$. In this paper some properties of the tensor centre of $G$ are obtained and the capability of the pair of groups $(G,G')$ is determined. Finally, the structure of $J_ {_2}(G)$ will be described, where $J_ {_2}(G)$ is the kernel of the map $\kappa: G\otimes G\rightarrow G^{'}$.

Keywords: non-abelian tensor square, tensor centre, relative central extension, capable group

MSC numbers: 20F99, 20F14