Editing Special linear group (section)

==SL<sup>±</sup>(''n'',''F'')==
In [[Characteristic (algebra)|characteristic]] other than 2, the set of matrices with determinant {{math|±1}} form another subgroup of GL, with SL as an index 2 subgroup (necessarily normal); in characteristic 2 this is the same as SL. This forms a [[short exact sequence]] of groups:
:<math>1\to\operatorname{SL}(n, F) \to \operatorname{SL}^{\pm}(n, F) \to \{\pm 1\}\to1.</math>

This sequence splits by taking any matrix with determinant {{math|−1}}, for example the diagonal matrix <math>(-1, 1, \dots, 1).</math> If <math>n = 2k + 1</math> is odd, the negative identity matrix <math>-I</math> is in {{math|SL<sup>±</sup>(''n'',''F'')}} but not in {{math|SL(''n'',''F'')}} and thus the group splits as an [[internal direct product]] <math>\operatorname{SL}^\pm(2k + 1, F) \cong \operatorname{SL}(2k + 1, F) \times \{\pm I\}</math>. However, if <math>n = 2k</math> is even, <math>-I</math> is already in {{math|SL(''n'',''F'')}} , {{math|SL<sup>±</sup>}} does not split, and in general is a non-trivial [[group extension]].

Over the real numbers, {{math|SL<sup>±</sup>(''n'', ''R'')}} has two [[connected component (topology)|connected components]], corresponding to {{math|SL(''n'', ''R'')}} and another component, which are isomorphic with identification depending on a choice of point (matrix with determinant {{math|−1}}). In odd dimension these are naturally identified by <math>-I</math>, but in even dimension there is no one natural identification.