On the vanishing of subspaces of alternating bilinear forms

Abstract

Given a field F and integer n≥3, we introduce an invariant sn (F) which is defined by examining the vanishing of subspaces of alternating bilinear forms on 2-dimensional subspaces of vector spaces. This invariant arises when we calculate the largest dimension of a subspace of n × n skew-symmetric matrices over F which contains no elements of rank 2. We show how to calculate sn (F) for various families of field F, including finite fields. We also prove the existence of large subgroups of the commutator subgroup of certain p-groups of class 2 which contain no non-identity commutators.