Together with the imaginary unit "i" and the antilinear map "k" := "ij", "j" equips "V" with the structure of a quaternionic vector space (i.e., "V" becomes a module over the division algebra of quaternions). From this point of view, quaternionic representation of a group "G" is a group homomorphism "φ": "G" → GL("V", H), the group of invertible quaternion-linear transformations of "V". In particular, a quaternionic matrix representation of "g" assigns a square matrix of quaternions "ρ"(g) to each element "g" of "G" such that "ρ"(e) is the identity matrix and