Special linear group is characteristic in general linear group

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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, Characteristic subgroup (?)) in a particular group or type of group (namely, General linear group (?)).

Statement

Suppose k is a field and n is a natural number. Let GL_n(k) denote the group of all invertible n \times n matrices over k, and SL_n(k) denote the subgroup comprising matrices of determinant 1. Then, Sl_n(k) is a characteristic subgroup of GL_n(k): every automorphism of GL_n(k) sends SL_n(k) to itself.

Related facts

Stronger facts

Facts used

  1. Commutator subgroup of general linear group is special linear group: This result holds except when n = 2 and k has two elements.
  2. Commutator subgroup is characteristic

Proof

The case of a field with two elements

In this case, every invertible matrix has determinant 1, because 1 is the only nonzero number in the field. Thus, SL_n(k) = GL_n(k). Since every group is a characteristic subgroup of itself, SL_n(k) is characteristic in GL_n(k).

Other cases

The proof follows from facts (1) and (2).