A homological invariant of certain torsion free crystallographic groups

Abstract

Several homological invariants namely the nonabelian tensor square, the exterior square and the Schur multiplier of groups have been of research interests by group theorists over the years. Besides, there are also some other homological invariants which can be deduced from these invariants, as example, the central subgroup of the nonabelian tensor square of a group G, known as ()G?. The computations of the homological invariants of crystallographic groups strengthen the link between group theory with crystallography theory. In this paper, ()G?is determined for certain torsion free crystallographic groups focusing on those with cyclic point groups of order three and five.