# Isomorphic general linear groups implies isomorphic fields

From Groupprops

## Statement

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

Note that the case is different, because is the multiplicative group of , and this does not determine up to isomorphism.

## Related facts

### Similar facts

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