  We study the subgroup structure of the semigroup of finitary tropical
matrices under multiplication. We show that every maximal subgroup is
isomorphic to the full linear automorphism group of a related tropical
polytope, and that each of these groups is the direct product of the real
numbers with a finite group. We also show that there is a natural and canonical
embedding of each full rank maximal subgroup into the group of units of the
semigroup of matrices over the tropical semiring with minus infinity. 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, and that every full rank subgroup has a
common eigenvector.
