Derived subgroup of general linear group is special linear group

From Groupprops
Jump to: navigation, search

Template:Sdf computation

Statement

Let k be a field and n be a natural number. The derived subgroup (i.e., commutator subgroup) of the general linear group GL_n(k) (the group of invertible n \times n matrices) is the special linear group SL_n(k) (the group of n \times n matrices of determinant 1), under either of these conditions:

  • n \ge 3.
  • k has at least three elements.

In other words, the only case where the result does not hold is when n = 2 and k is the field of two elements. (In the case n = 1, the result holds vacuously).

Related facts

Facts used

  1. Every elementary matrix is a commutator of invertible matrices
  2. Elementary matrices generate the special linear group

Proof

Observe that:

  • SL_n(k) is the kernel of the determinant homomorphism from GL_n(k) to the multiplicative group of nonzero elements of k, which is Abelian. Hence, SL_n(k) contains the commutator subgroup of GL_n(k).
  • By facts (1) and (2), SL_n(k) is contained in the commutator subgroup of GL_n(k).
  • Thus, SL_n(k) equals the commutator subgroup of GL_n(k).