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

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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 k be a field and n be a natural number. Let GL_n(k) be the group of invertible n \times n mtarices over k, and SL_n(k) be the special linear group: the subgroup comprising matrices of determinant one.

If the map x \mapsto x^n is a surjective map from k to k, then SL_n(k) is a cocentral subgroup of GL_n(k): its product with the center of GL_n(k) equals GL_n(k). In particular:

  • For k a finite field of order q, the n^{th} power map is surjective if and only if n is relatively prime to q - 1.
  • For k = \R, the n^{th} power map is surjective if and only if n is odd.
  • For k an algebraically closed field, the n^{th} power map is always surjective.

Related facts

Corollaries

Other related facts

Facts used

  1. center of general linear group is group of scalar matrices over center

Proof

Since SL_n(k) is the kernel of the determinant homomorphism, it suffices to show that the image of the center of GL_n(k), under the determinant homomorphism, equals the whole of k^*.

By fact (1), the center of GL_n(k) is the group of scalar matrices with scalar entry a nonzero element of k. If this nonzero element is \lambda, the determinant is \lambda^n. Thus, the map to k^* is surjective if and only if every nonzero element is the n^{th} power of some nonzero element, which happens if and only if the n^{th} power map is surjective.