Isomorphic general linear groups implies isomorphic fields

Statement
Suppose $$K_1,K_2$$ are fields and $$n > 1$$ is a natural number. Then, if the general linear groups $$GL(n,K_1)$$ and $$GL(n,K_2)$$ are isomorphic groups, $$K_1$$ and $$K_2$$ must be isomorphic as fields.

Note that the case $$n = 1$$ is different, because $$GL(1,K)$$ is the multiplicative group of $$K$$, and this does not determine $$K$$ up to isomorphism.

Similar facts

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

Opposite facts

 * Isomorphic general linear groups not implies isomorphic rings