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 (?)).
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.
- Projective special linear group equals projective general linear group iff nth power map is surjective
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.