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Establishing a distance between genomes is a significant problem in computational genomics, because its solution can be used to establish evolutionary relationships including phylogeny. The “double cut and join” (DCJ) model of chromosomal rearrangement proposed by Yancopoulos et al. (Bioinformatics 21:3340–3346, 2005) has received attention as it can model inversions, translocations, fusion and fission on a multichromosomal genome that may contain both linear and circular chromosomes. In this paper, we realize the DCJ operator as a group action on the space of multichromosomal genomes. We study this group action, deriving some properties of the group and finding group-theoretic analogues for the key results in the DCJ theory.