A nonstandard representation of PE₃(ℤ) by rational permutation matrices

Abstract

Reduction modulo two and the action on the seven nonzero vectors of $\mathbb F_2^3$ define a homomorphism $PE_3(\mathbb Z)\longrightarrow\operatorname{GL}_7(\mathbb Q)$, where $PE_3(\mathbb Z)$ is the image of the elementary group in the ambient projective general linear group. We prove that every elementary Weyl element has a nonidentity involution as its image. Such an image cannot arise from the signed Weyl matrix over the endomorphism ring of any rational module through an injective group homomorphism and a change of module coordinates. The resulting representation is therefore not induced by a standard homomorphism. This supplies an explicit particular case outside the condition in Kourovka Problem 10.43.