Special linear group is fully 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, Fully characteristic 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) denote the group of invertible n \times n matrices over k under multiplication and SL_n(k) denote the subgroup comprising matrices of determinant one. SL_n(k) is a fully characteristic subgroup of GL_n(k): every endomorphism of GL_n(k) restricts to an endomorphism of SL_n(k).

Facts used

  1. Commutator subgroup of general linear group is special linear group: This is true except in the case where n = 2 and the field k has exactly two elements.
  2. Commutator subgroup is fully characteristic

Proof

For the field with two elements

In this case, every element of GL_n(k) has determinant one, so SL_n(k) = GL_n(k). Since every group is fully characteristic as a subgroup of itself, SL_n(k) is a fully characteristic subgroup of GL_n(k).

For other cases

In all other cases, the result follows from facts (1) and (2).