On the non abelian tensor product of a group and its central automorphisms

The non-abelian tensor product of groups has it's origin in K-algebraic theory and topology and was first introduced by R. Brown and J.L. Loday in 1987 .One of the first topics which was studied on G⊗G is that whether the properties of G⊗G inherited from G or not? For instance, Bacon in 1994 determined an upper bound for the number of minimal generators of G⊗G in terms of the number of minimal generators of G.Let G be a group and Autz (G) be the group of It's central automorphisms, which is a normal subgroup of Aut(G). Our goal is to obtain an estimate for the number of minimal generators of G⊗Autz(G). For this, we first identify it's minimal generators. Then, when both G and Autz(G) are nilpotent groups of class two, we give an upper bound for d(G⊗Autz(G)) in terms of d(G) and d(Autz(G)), where d(X) is the minimal number of generators of X.