# Special linear group is cocentral in general linear group iff nth power map is surjective

This article gives the statement, and possibly proof, of a particular subgroup or type of subgroup (namely, Special linear group (?)) satisfying a particular subgroup property (namely, Cocentral subgroup (?)) in a particular group or type of group (namely, General linear group (?)).

## Statement

Let be a field and be a natural number. Let be the group of invertible mtarices over , and be the special linear group: the subgroup comprising matrices of determinant one.

If the map is a surjective map from to , then is a cocentral subgroup of : its product with the center of equals . In particular:

- For a finite field of order , the power map is surjective if and only if is relatively prime to .
- For , the power map is surjective if and only if is odd.
- For an algebraically closed field, the power map is always surjective.

## Related facts

### Corollaries

## Facts used

## Proof

Since is the kernel of the determinant homomorphism, it suffices to show that the image of the center of , under the determinant homomorphism, equals the whole of .

By fact (1), the center of is the group of scalar matrices with scalar entry a nonzero element of . If this nonzero element is , the determinant is . Thus, the map to is surjective if and only if every nonzero element is the power of some nonzero element, which happens if and only if the power map is surjective.