Special linear group is perfect

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

Statement

Let k be any field. Consider the special linear group SL_n(k): the group of n \times n matrices over k that have determinant 1. Then, the following are true:

  • For n \ge 3, SL_n(k) is a perfect group for any field k.
  • For n = 2, SL_n(k) is a perfect group if k has more than three elements.

Related facts

Facts used

  1. Elementary matrices generate the special linear group
  2. Every elementary matrix is a commutator of unimodular matrices: If n \ge 3 or n = 2 and k has more than three elements, every elementary n \times n matrix is a commutator of two elements of SL_n(k).

Proof

The proof follows by piecing together facts (1) and (2).