The computation of the nonabelian tensor product of cyclic groups of order p2

Abstract

Let G and H be groups which act on each other and each of which acts on itself by conjugation, then the actions are compatible if (gh)g' = g(h(g-1 g')) and (hg)h' = h(g(h-1 h')) for g,g'∈G and h,h'∈H. Compatible actions play a very important role in determining the nonabelian tensor product. The nonabelian tensor product, G⊗H, was introduced by Brown and Loday in 1984. The nonabelian tensor product is the group generated by g⊗h with two relations gg'⊗h = (gg'⊗gh)(g⊗h) and g⊗hh' = (g⊗h)(hg⊗hh') for g,g'∈G and h,h'∈H, where G and H act on each other in a compatible fashion and act on themselves by conjugation. In 1987, Brown et al. gave an open problem in determining whether the tensor product of two cyclic groups is cyclic. Visscher in 1998 has shown that the nonabelian tensor product is not necessarily cyclic, but he only focused on the case of cyclic groups of 2-power order where the action is of order two. In this paper, the compatibility and the nonabelian tensor product of cyclic groups of order p2 with the actions of order p are determined.