Isomorphic unitriangular matrix groups implies isomorphic fields

Statement
Suppose $$K_1,K_2$$ are fields and $$n > 2$$ is a natural number. Then, if the unitriangular matrix groups $$UT(n,K_1)$$ and $$UT(n,K_2)$$ are isomorphic groups, $$K_1$$ and $$K_2$$ must be isomorphic as fields.

Note that the cases $$n = 1$$ and $$n = 2$$ are different. $$UT(1,K)$$ is the trivial group and $$UT(2,K)$$ is isomorphic to the additive group of $$K$$, which does not determine $$K$$ as a field (for instance, any two quadratic extensions of the rationals have isomorphic additive groups).

Related facts

 * Isomorphic general linear groups implies isomorphic fields
 * Isomorphic special linear groups implies isomorphic fields
 * Isomorphic general affine groups implies isomorphic fields