NON-ABELIAN TENSOR ABSOLUTE CENTRE OF A GROUP

Abstract

Abstract. 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 $K^\otimes(G)=G\otimes {\rm Aut}(G)$. Then under some condition we prove that the finiteness of $G/L^\otimes(G)$ implies that $K^\otimes(G)$ is also finite.