Exterior products of torsion free Abelian groups and nilpotent groups of class two. (Q1073908)

The author investigates the structure of the exterior product \(H\wedge H\) where H is a torsion free abelian group of finite rank. If H has rank n, then H and \(H\wedge H\) correspond in a natural way to rational projective spaces of dimensions n-1 and \(\left( \begin{matrix} n\\ 2\end{matrix} \right)-1\) respectively. Using Grassmannian algebra and Plücker identities the special cases of \(n=3\) and \(n=4\) are discussed in detail. Finally, the earlier structural theorems are applied to torsionfree nilpotent groups G of class 2 with G/G' torsion free of rank n and G' torsionfree of rank \(\left( \begin{matrix} n\\ 2\end{matrix} \right)\) since in this case G'\(\simeq G/G'\wedge G/G'\). In particular, it is shown that every element of G' is a commutator if G/G' and G' both have rank 3.