Editing Commutator subgroup (section)

== Commutators ==
{{main|Commutator}}
For elements <math>g</math> and <math>h</math> of a group ''G'', the [[commutator]] of <math>g</math> and <math>h</math> is <math>[g,h] = g^{-1}h^{-1}gh</math>.  The commutator <math>[g,h]</math> is equal to the [[identity element]] ''e'' if and only if <math>gh = hg</math> , that is, if and only if <math>g</math> and <math>h</math> commute.  In general, <math>gh = hg[g,h]</math>.

However, the notation is somewhat arbitrary and there is a non-equivalent variant definition for the commutator that has the inverses on the right hand side of the equation: <math>[g,h] = ghg^{-1}h^{-1}</math> in which case  <math>gh \neq hg[g,h]</math> but instead <math>gh = [g,h]hg</math>.

An element of ''G'' of the form <math>[g,h]</math> for some ''g'' and ''h'' is called a commutator.  The identity element ''e'' = [''e'',''e''] is always a commutator, and it is the only commutator if and only if ''G'' is abelian.

Here are some simple but useful commutator identities, true for any elements ''s'', ''g'', ''h'' of a group ''G'':

* <math>[g,h]^{-1} = [h,g],</math>
* <math>[g,h]^s = [g^s,h^s],</math> where <math>g^s = s^{-1}gs</math> (or, respectively, <math> g^s = sgs^{-1}</math>) is the [[Conjugacy class|conjugate]] of <math>g</math> by <math>s,</math>
* for any [[Group homomorphism|homomorphism]] <math>f: G \to H </math>, <math>f([g, h]) = [f(g), f(h)].</math>

The first and second identities imply that the [[Set (mathematics)|set]] of commutators in ''G'' is closed under inversion and conjugation.  If in the third identity we take ''H'' = ''G'', we get that the set of commutators is stable under any [[endomorphism]] of ''G''.  This is in fact a generalization of the second identity, since we can take ''f'' to be the conjugation [[automorphism]] on ''G'', <math> x \mapsto x^s </math>, to get the second identity.

However, the product of two or more commutators need not be a commutator.  A generic example is [''a'',''b''][''c'',''d''] in the [[free group]] on ''a'',''b'',''c'',''d''.  It is known that the least order of a finite group for which there exists two commutators whose product is not a commutator is 96; in fact there are two nonisomorphic groups of order 96 with this property.<ref>{{harvtxt|Suárez-Alvarez}}</ref>