Isomorphic unitriangular matrix groups implies isomorphic fields

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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).

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