On certain Abelian groups associated with finite projective geometries (Q910711)

Let p be a prime and \(n\in N\setminus \{1\}\). By \(P_{n-1}(p)\) is denoted the projective space over the finite field Fp. There are 3 finite Abelian groups naturally associated with \(P_{n-1}(p)\). One of them is isomorphic to a vector space of dimension n over Fp. The other two are factor groups of the free Abelian group \({\mathbb{Z}}^ k\) with \(k=(p^ n- 1)/(p-1)\); they are defined by the rows of the point-hyperplane incidence matrix A of \(P_{n-1}(p)\) and the complement \(A'\) of A, respectively. These groups are denoted by H(A) and \(H(A').\) The main result of the paper is the following Theorem: (a) The invariant factors of \(A'\) are the integers \(p^{i-1}\) with multiplicity \(d_ i\), \(i=1,...,n\), where \(d_ i\) is the coefficient of \(\lambda^{i(p-1)}\) in the expansion of \((\sum^{p- 1}_{j=0}\lambda^ j)^ n\). (b) The invariant factors of A are the integers: (i) 1 with multiplicity \(d_ 1+1\), (ii) \(p^{i-1}\) with multiplicity \(d_ i\), \(i=2,...,n-2\), (iii) \(p^{n-2}\) with multiplicity \(d_{n-1}-1\), and \((iv)\quad p^{n-2}(p^{n-1}-1)/(p-1)\) with multiplicity 1, where the \(d_ i\) are as in (a).