The Exterior Squares of Some Crystallographic Groups

A crystallographic group is a discrete subgroup G of the set of isometries of Euclidean space En, where the quotient space En/G is compact. A specific type of crystallographic groups is called Bieberbach groups. A Bieberbach group is defined to be a torsion free crystallographic group. In this paper, the exterior squares of some Bieberbach groups with abelian point groups are computed. The exterior square of a group is the factor group of the nonabelian tensor square with the central subgroup of the group.