Semantic editing of images is a fundamental goal of computer vision. While generative adversarial networks (GANs) are gaining attention for their ability to produce high-quality images, they do not provide an inherent way to edit images semantically. Recent studies have investigated how to manipulate the latent variable to determine the images to be generated. However, methods that assume linear semantic arithmetic have limitations in the quality of image editing. Also, methods that discover nonlinear semantic pathways provide editing that is non-commutative, in other words, inconsistent when applied in different orders. This paper proposes a method for discovering semantic commutative vector fields. We theoretically demonstrate that thanks to commutativity, multiple editing along the vector fields depend only on the quantities of editing, not on the order of the editing. We also experimentally demonstrated that the nonlinear and commutative nature of editing provides higher quality editing than previous methods.