Advances in Computing the Non-abelian Tensor Square of Polycyclic Groups

Abstract

The nonabelian tensor square G ⊗ G of the group G is the group generated by the symbols g ⊗ h, where g, h ∈ G, subject to the relations gg ′ ⊗ h = ( g g ′ ⊗ g h)(g ⊗ h) and g ⊗ hh ′ = (g ⊗ h) ( h g ⊗ h h ′) for all g, g, h, h ′ ∈ G, where g g ′ = gg ′ g −1 is conjugation on the left. Following the work of C. Miller [18], R. K. Dennis in [10] introduced the nonabelian tensor square which is a specialization of the more general nonabelian tensor product independently introduced by R. Brown and J.-L. Loday [6]. By computing the nonabelian tensor square we mean finding a standard or simplified presentation for it. In the case of finite groups, the definition of the nonabelian tensor square gives a finite presentation that can be simplified using Tietze transformations. This simplified presentation can then be examined to determine the nonabelian tensor square. This was the approach taken in [3], in which the nonabelian tensor square was computed for each nonabelian group of order up to 30. Creating a presentation from the definition of the nonabelian tensor square, simplifying it using Tietze transformations and computing a structure description from the simplified presentation can be implemented in few lines of GAP [16]. However, this strategy does not scale well to finite groups G having order greater than 100 since the initial presentation has |G | 2 generators and 2|G | 3 relations. The most general method for computing the nonabelian tensor square uses the notion of a crossed pairing (see [3]). Let G and L be