Editing Commutator (section)

=== Identities (group theory) ===
Commutator identities are an important tool in [[group theory]].<ref>{{harvtxt|McKay|2000|p=4}}</ref> The expression {{math|''a<sup>x</sup>''}} denotes the [[conjugate (group theory)#Definition|conjugate]] of {{mvar|a}} by {{mvar|x}}, defined as {{math|''x''<sup>−1</sup>''ax''}}.
# <math>x^y = x^{-1}[x, y].</math>
# <math>[y, x] = [x,y]^{-1}.</math>
# <math>[x, zy] = [x, y]\cdot [x, z]^y</math> and <math>[x z, y] = [x, y]^z \cdot [z, y].</math>
# <math>\left[x, y^{-1}\right] = [y, x]^{y^{-1}}</math> and <math>\left[x^{-1}, y\right] = [y, x]^{x^{-1}}.</math>
# <math>\left[\left[x, y^{-1}\right], z\right]^y \cdot \left[\left[y, z^{-1}\right], x\right]^z \cdot \left[\left[z, x^{-1}\right], y\right]^x = 1</math> and <math>\left[\left[x, y\right], z^x\right] \cdot  \left[[z ,x], y^z\right] \cdot \left[[y, z], x^y\right] = 1.</math>

Identity (5) is also known as the ''Hall–Witt identity'', after [[Philip Hall]] and [[Ernst Witt]].  It is a group-theoretic analogue of the [[Jacobi identity]] for the ring-theoretic commutator (see next section).

N.B., the above definition of the conjugate of {{mvar|a}} by {{mvar|x}} is used by some group theorists.<ref>{{harvtxt|Herstein|1975|p=83}}</ref>  Many other group theorists define the conjugate of {{mvar|a}} by {{mvar|x}} as {{math|''xax''<sup>−1</sup>}}.<ref>{{harvtxt|Fraleigh|1976|p=128}}</ref>  This is often written <math>{}^x a</math>.  Similar identities hold for these conventions.

Many identities that are true modulo certain subgroups are also used.  These can be particularly useful in the study of [[solvable group]]s and [[nilpotent group]]s.  For instance, in any group, second powers behave well:
: <math>(xy)^2 = x^2 y^2 [y, x][[y, x], y].</math>

If the [[derived subgroup]] is central, then
: <math>(xy)^n = x^n y^n [y, x]^\binom{n}{2}.</math>