Covering [formula omitted] and products of blockcyclic matrices

Abstract

Let V be an n-dimensional vector space over a field K (where n∈N,n⩾3,∣K∣⩾4). A linear transformation φ:V→V is called blockcyclic if the minimum rank satisfies δ(φ)⩾n/2. We study products of blockcyclic conjugacy classes in GL(V). Example: If Ω and Φ are products of blockcyclic conjugacy classes in GL(V) then ΩΦ contains a cyclic transformation. The results are used in our study on products of singular similarity classes (Knüppel and Nielsen, in press) [8]. A further application of our methods is the following theorem. Let k∈N⩾n+1, Ω a non-central conjugacy class of GL(V) and π∈GL(V) such that det(π)=det(Ωk). Then π∈Ωk. A similar result was proved in Lev’s article (Lev, 1996) [10].