# Isomorphic unitriangular matrix groups implies isomorphic fields

From Groupprops

## Statement

Suppose are fields and is a natural number. Then, if the unitriangular matrix groups and are isomorphic groups, and must be isomorphic as fields.

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