Commutator subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
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For elements g {\displaystyle g} and h {\displaystyle h} of a group G, the commutator of g {\displaystyle g} and h {\displaystyle h} is [ g , h ] = g − 1 h − 1 g h {\displaystyle [g,h]=g^{-1}h^{-1}gh} . The commutator [ g , h ] {\displaystyle [g,h]} is equal to the identity element e if and only if g h = h g {\displaystyle gh=hg} , that is, if and only if g {\displaystyle g} and h {\displaystyle h} commute. In general, g h = h g [ g , h ] {\displaystyle gh=hg[g,h]} . 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: [ g , h ] = g h g − 1 h − 1 {\displaystyle [g,h]=ghg^{-1}h^{-1}} in which case g h ≠ h g [ g , h ] {\displaystyle gh\neq hg[g,h]} but instead g h = [ g , h ] h g {\displaystyle gh=[g,h]hg} . An element of G of the form [ g , h ] {\displaystyle [g,h]} 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.
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This motivates the definition of the commutator subgroup [ G , G ] {\displaystyle [G,G]} (also called the derived subgroup, and denoted G ′ {\displaystyle G'} or G ( 1 ) {\displaystyle G^{(1)}} ) of G: it is the subgroup generated by all the commutators. It follows from this definition that any element of [ G , G ] {\displaystyle [G,G]} is of the form for some natural number n {\displaystyle n} , where the gi and hi are elements of G. Moreover, since ( [ g 1 , h 1 ] ⋯ [ g n , h n ] ) s = [ g 1 s , h 1 s ] ⋯ [ g n s , h n s ] {\displaystyle ([g_{1},h_{1}]\cdots [g_{n},h_{n}])^{s}=[g_{1}^{s},h_{1}^{s}]\cdots [g_{n}^{s},h_{n}^{s}]} , the commutator subgroup is normal in G. For any homomorphism f: G → H, so that f ( [ G , G ] ) ⊆ [ H , H ] {\displaystyle f([G,G])\subseteq [H,H]} . This shows that the commutator subgroup can be viewed as a functor on the category of groups, some implications of which are explored below. Moreover, taking G = H it shows that the commutator subgroup is stable under every endomorphism of G: that is, [G,G] is a fully characteristic subgroup of G, a property considerably stronger than normality.
Derived series
The groups G ( 2 ) , G ( 3 ) , … {\displaystyle G^{(2)},G^{(3)},\ldots } are called the second derived subgroup, third derived subgroup, and so forth, and the descending normal series is called the derived series. This should not be confused with the lower central series, whose terms are G n := [ G n − 1 , G ] {\displaystyle G_{n}:=[G_{n-1},G]} . For a finite group, the derived series terminates in a perfect group, which may or may not be trivial. For an infinite group, the derived series need not terminate at a finite stage, and one can continue it to infinite ordinal numbers via transfinite recursion, thereby obtaining the transfinite derived series, which eventually terminates at the perfect core of the group.
Abelianization
Given a group G {\displaystyle G} , a quotient group G / N {\displaystyle G/N} is abelian if and only if [ G , G ] ⊆ N {\displaystyle [G,G]\subseteq N} . The quotient G / [ G , G ] {\displaystyle G/[G,G]} is an abelian group called the abelianization of G {\displaystyle G} or G {\displaystyle G} made abelian. It is usually denoted by G ab {\displaystyle G^{\operatorname {ab} }} or G ab {\displaystyle G_{\operatorname {ab} }} . There is a useful categorical interpretation of the map φ : G → G ab {\displaystyle \varphi :G\rightarrow G^{\operatorname {ab} }} . Namely φ {\displaystyle \varphi } is universal for homomorphisms from G {\displaystyle G} to an abelian group H {\displaystyle H} : for any abelian group H {\displaystyle H} and homomorphism of groups f : G → H {\displaystyle f:G\to H} there exists a unique homomorphism F : G ab → H {\displaystyle F:G^{\operatorname {ab} }\to H} such that f = F ∘ φ {\displaystyle f=F\circ \varphi } . As usual for objects defined by universal mapping properties, this shows the uniqueness of the abelianization G ab {\displaystyle G^{\operatorname {ab} }} up to canonical isomorphism, whereas the explicit construction G → G / [ G , G ] {\displaystyle G\to G/[G,G]} shows existence.
Classes of groups
A group G {\displaystyle G} is an abelian group if and only if the derived group is trivial: [G,G] = {e}. Equivalently, if and only if the group equals its abelianization. See above for the definition of a group's abelianization. A group G {\displaystyle G} is a perfect group if and only if the derived group equals the group itself: [G,G] = G. Equivalently, if and only if the abelianization of the group is trivial. This is, in a sense, the opposite of being abelian. A group with G ( n ) = { e } {\displaystyle G^{(n)}=\{e\}} for some n in N is called a solvable group; this is weaker than abelian, which is the case n = 1. A group with G ( n ) ≠ { e } {\displaystyle G^{(n)}\neq \{e\}} for all n in N is called a non-solvable group.
Perfect group
Whenever a group G {\displaystyle G} has derived subgroup equal to itself, G ( 1 ) = G {\displaystyle G^{(1)}=G} , it is called a perfect group. This includes non-abelian simple groups and the special linear groups SL n ( k ) {\displaystyle \operatorname {SL} _{n}(k)} for a fixed field k {\displaystyle k} .
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Map from Out
Since the derived subgroup is characteristic, any automorphism of G induces an automorphism of the abelianization. Since the abelianization is abelian, inner automorphisms act trivially, hence this yields a map
