Editing Projective linear group (section)

=== Order ===
The order of {{nowrap|PGL(''n'', ''q'')}} is
: (''q''<sup>''n''</sup> − 1)(''q<sup>n</sup>'' − ''q'')(''q<sup>n</sup>'' − ''q''<sup>2</sup>) ⋅⋅⋅ (''q''<sup>''n''</sup> − ''q''<sup>''n''−1</sup>)/(gcd(''n'', ''q'' − 1)) = ''q''<sup>''n''<sup>2</sup>−1</sup> − O(''q''<sup>''n''<sup>2</sup>−3</sup>),
which corresponds to the [[General linear group#Over finite fields|order of {{nowrap|GL(''n'', ''q'')}}]], divided by {{nowrap|''q'' − 1}} for projectivization; see [[q-analog|''q''-analog]] for discussion of such formulas. Note that the degree is {{nowrap|''n''<sup>2</sup> − 1}}, which agrees with the dimension as an algebraic group. The "O" is for [[big O notation]], meaning "terms involving lower order". This also equals the order of {{nowrap|SL(''n'', ''q'')}}; there dividing by {{nowrap|''q'' − 1}} is due to the determinant.

The order of {{nowrap|PSL(''n'', ''q'')}} is the order of {{nowrap|PGL(''n'', ''q'')}} as above, divided by {{nowrap|gcd(''n'', ''q'' − 1)}}. This is equal to {{nowrap|{{abs|SZ(''n'', ''q'')}}}}, the number of scalar matrices with determinant 1; {{abs|''F''<sup>×</sup>{{hsp}}/{{hsp}}(''F''<sup>×</sup>)<sup>''n''</sup>}}, the number of classes of element that have no ''n''th root; and it is also the number of ''n''th [[roots of unity]] in '''F'''<sub>''q''</sub>.<ref group="note">These are equal because they are the kernel and cokernel of the endomorphism {{nowrap|''F''<sup>×</sup> {{overset|lh=0.6|''x''<sup>''n''</sup>|→}} ''F''<sup>×</sup>}}; formally, {{nowrap|1={{abs|''μ''<sub>''n''</sub>}} ⋅ {{abs|(''F''<sup>×</sup>)<sup>''n''</sup>}} = {{abs|''F''<sup>×</sup>}}}}. More abstractly, the first realizes PSL as SL{{hsp}}/{{hsp}}SZ, while the second realizes PSL as the kernel of {{nowrap|PGL → ''F''<sup>×</sup>{{hsp}}/{{hsp}}(''F''<sup>×</sup>)<sup>''n''</sup>}}.</ref>