Tropical matrix groups

We study the structure of groups of finitary tropical matrices under multiplication. We show that the maximal groups of \(n \times n\) tropical matrices are precisely the groups of the form \(G \times \mathbb {R}\) where G is a group admitting a 2-closed permutation representation on n points. Each such maximal group is also naturally isomorphic to the full linear automorphism group of a related tropical polytope. Our results have numerous corollaries, including the fact that every automorphism of a projective (as a module) tropical polytope of full rank extends to an automorphism of the containing space.