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Let $X$ be an abstract tropical curve and let $G$ be a finite subgroup of theautomorphism group of $X$. Let $D$ be a divisor on $X$ whose equivalence classis $G$-invariant. We address the following question: is there always a divisor$D'$ in the equivalence class of $D$ which is $G$-invariant? Our main result isthat the answer is "yes" for all abstract tropical curves. A key step in ourproof is a tropical analogue of Hilbert's Theorem 90.

Automorphism Groups on Tropical Curves: Some Cohomology Calculations