NON-ABELIAN TENSOR ABSOLUTE CENTRE OF A GROUP

In 1904, schur proved his famous result which says that if the central factor group of a given group is finite then so is its derived subgroup. in 1994, hegarty showed that if the absolute central factor group, g/l(g), is finite then so is its autocommutator subgroup, k(g). using the notion of non-abelian tensor product, we introduce the concept of tensor absolute centre, ‎$l^otimes (g)$, and $k^otimes(g)=gotimes {rm aut}(g)$. then under some condition we prove that the finiteness of $g/l^otimes(g)$ implies that $k^otimes(g)$ is also finite.