Editing Permutation matrix (section)

== Matrix group ==
When {{pi}} is the identity permutation, which has <math>\pi(i)=i</math> for all ''i'', both {{math|''C''<sub>{{pi}}</sub>}} and {{math|''R''<sub>{{pi}}</sub>}} are the [[identity matrix]].

There are {{math|''n''!}} permutation matrices, since there are {{math|''n''!}} permutations and the map <math>C\colon\pi\mapsto C_\pi</math> is a one-to-one correspondence between permutations and permutation matrices.  (The map <math>R</math> is another such correspondence.)  By the formulas above, those {{math|''n'' × ''n''}} permutation matrices form a [[Group (mathematics)|group]] of order {{math|''n''!}} under matrix multiplication, with the identity matrix as its [[identity element]], a group that we denote <math>\mathcal{P}_n</math>.  The group <math>\mathcal{P}_n</math> is a subgroup of the [[general linear group]] <math>GL_n(\mathbb{R})</math> of invertible {{math|''n'' × ''n''}} matrices of real numbers.  Indeed, for any [[Field (mathematics)|field]] ''F'', the group  <math>\mathcal{P}_n</math> is also a subgroup of the group <math>GL_n(F)</math>, where the matrix entries belong to ''F''. (Every field contains 0 and 1 with <math>0+0=0,</math> <math>0+1=1,</math> <math>0*0=0,</math> <math>0*1=0,</math> and <math>1*1=1;</math> and that's all we need to multiply permutation matrices.  Different fields disagree about whether <math>1+1=0</math>, but that sum doesn't arise.)

Let <math>S_n^\leftarrow</math> denote the [[symmetric group]], or [[permutation group|group of permutations]], on {1,2,...,{{math|''n''}}} where the group operation is the standard, right-to-left composition "<math>\circ</math>"; and let <math>S_n^\rightarrow</math> denote the [[opposite group]], which uses the left-to-right composition "<math>\,;\,</math>".  The map <math>C\colon S_n^\leftarrow\to GL_n(\mathbb{R})</math> that takes {{pi}} to its column-based matrix <math>C_\pi</math> is a [[faithful representation]], and similarly for the map <math>R\colon S_n^\rightarrow\to GL_n(\mathbb{R})</math> that takes {{pi}} to <math>R_\pi</math>.