The present paper is devoted to investigation of the isometry group of the Gromov-Hausdorff space, i.e., the metric space of compact metric spaces considered up to isometry and endowed with the Gromov-Hausdorff metric. The main goal is to present a complete proof of the following result by G. Lowther (2015): the isometry group of the Gromov-Hausdorff space is trivial.