We prove that the class of nilpotent by finite, solvable by finite, polycyclic by finite, nilpotent of nilpotency class n and supersolvable groups are closed under the formation of the non-abelian tensor product. We provide necessary and sufficient conditions for the non-abelian tensor product of finitely generated groups to be finitely generated. We prove that central extensions of most finite simple groups have trivial Bogomolov multiplier.

Abstract. In this note, we give a homology-free proof that the non-abelian tensor product of two finite groups is finite. In addition, we provide an explicit proof that the non-abelian tensor product of two finite p-groups is a finite p-group.

We prove that for a finitely generated group G, the second stable homotopy

In [3], Bazzoni and Glaz conjecture that the weak global dimension of a Gaussian ring is 0, 1 or ∞. In this paper, we prove their conjecture.2010 Mathematics Subject Classification. Primary 13B25,13D05,13F05,16E30 Secondary 16E65.

The purpose of this paper is two fold. First we introduce the box-tensor product of two groups as a generalization of the nonabelian tensor product of groups. We extend various results for nonabelian tensor products to the box-tensor product such as the finiteness of the product when each factor is finite. This would give yet another proof of Ellis's theorem on the finiteness of the nonabelian tensor product of groups when each factor is finite. Secondly, using the methods developed in proving the finiteness of the box-tensor product, we prove the finiteness of Inassaridze's tensor product under some additional hypothesis which generalizes his results on the finiteness of his product. In addition, we prove an Ellis like finiteness theorem under weaker assumptions, which is a generalization of his theorem on the finiteness of nonabelian tensor product. As a consequence, we prove the finiteness of low-dimensional nonabelian homology groups.