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 2 G, subject to the relations gg 0 › h = ( g g 0 › g h)(g › h) and g › hh 0 = (g › h)( h g › h h 0 ) for all g;g;h;h 0 2 G, where g g 0 = gg 0 g i1 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 jGj 2 generators and 2jGj 3 relations.