Editing Commutator subgroup (section)

== Definition ==
This motivates the definition of the '''commutator subgroup''' <math>[G, G]</math> (also called the '''derived subgroup''', and denoted <math>G'</math> or <math>G^{(1)}</math>) of ''G'': it is the subgroup [[generating set of a group|generated]] by all the commutators.

It follows from this definition that any element of <math>[G, G]</math> is of the form

:<math>[g_1,h_1] \cdots [g_n,h_n] </math>

for some [[natural number]] <math>n</math>, where the ''g''<sub>''i''</sub> and ''h''<sub>''i''</sub> are elements of ''G''.  Moreover, since <math>([g_1,h_1] \cdots [g_n,h_n])^s = [g_1^s,h_1^s] \cdots [g_n^s,h_n^s]</math>, the commutator subgroup is normal in ''G''.  For any homomorphism ''f'': ''G'' → ''H'',

:<math>f([g_1,h_1] \cdots [g_n,h_n]) = [f(g_1),f(h_1)] \cdots [f(g_n),f(h_n)]</math>,

so that <math>f([G,G]) \subseteq [H,H]</math>.

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.

The commutator subgroup can also be defined as the set of elements ''g'' of the group that have an expression as a product ''g'' = ''g''<sub>1</sub> ''g''<sub>2</sub> ... ''g''<sub>''k''</sub> that can be rearranged to give the identity.

=== Derived series ===
This construction can be iterated:
:<math>G^{(0)} := G</math>
:<math>G^{(n)} := [G^{(n-1)},G^{(n-1)}] \quad n \in \mathbf{N}</math>
The groups <math>G^{(2)}, G^{(3)}, \ldots</math> are called the '''second derived subgroup''', '''third derived subgroup''', and so forth, and the descending [[normal series]]
:<math>\cdots \triangleleft G^{(2)} \triangleleft G^{(1)} \triangleleft G^{(0)} = G</math>
is called the '''derived series'''. This should not be confused with the '''[[lower central series]]''', whose terms are <math>G_n := [G_{n-1},G]</math>.

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 number]]s via [[transfinite recursion]], thereby obtaining the '''transfinite derived series''', which eventually terminates at the [[perfect core]] of the group.

=== Abelianization ===
Given a group <math>G</math>, a [[quotient group]] <math>G/N</math> is abelian if and only if <math>[G, G]\subseteq N</math>.

The quotient <math>G/[G, G]</math> is an abelian group called the '''abelianization''' of <math>G</math> or <math>G</math> '''made abelian'''.<ref>{{harvtxt|Fraleigh|1976|p=108}}</ref> It is usually denoted by <math>G^{\operatorname{ab}}</math> or <math>G_{\operatorname{ab}}</math>.

There is a useful categorical interpretation of the map <math>\varphi: G \rightarrow G^{\operatorname{ab}}</math>.  Namely <math>\varphi</math> is universal for homomorphisms from <math>G</math> to an abelian group <math>H</math>: for any abelian group <math>H</math> and homomorphism of groups <math>f: G \to H</math> there exists a unique homomorphism <math>F: G^{\operatorname{ab}}\to H</math> such that <math>f = F \circ \varphi</math>.  As usual for objects defined by universal mapping properties, this shows the uniqueness of the abelianization <math>G^{\operatorname{ab}}</math> up to canonical isomorphism, whereas the explicit construction <math>G\to G/[G, G]</math> shows existence.

The abelianization functor is the [[adjoint functors|left adjoint]] of the inclusion functor from the [[category of abelian groups]] to the category of groups. The existence of the abelianization functor '''Grp''' → '''Ab''' makes the category '''Ab''' a [[reflective subcategory]] of the category of groups, defined as a full subcategory whose inclusion functor has a left adjoint. 

Another important interpretation of <math>G^{\operatorname{ab}}</math> is as <math>H_1(G, \mathbb{Z})</math>, the first [[group homology|homology group]] of <math>G</math> with integral coefficients.

=== Classes of groups ===
A group <math>G</math> 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 <math>G</math> 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 "opposite" to abelian.

A group with <math>G^{(n)}=\{e\}</math> for some ''n'' in '''N''' is called a '''[[solvable group]]'''; this is weaker than abelian, which is the case ''n'' = 1.

A group with <math>G^{(n)} \neq \{e\}</math> for all ''n'' in '''N''' is called a '''non-solvable group'''.

A group with <math>G^{(\alpha)}=\{e\}</math> for some [[ordinal number]], possibly infinite, is called a '''[[perfect radical|hypoabelian group]]'''; this is weaker than solvable, which is the case ''α'' is finite (a natural number).

=== Perfect group ===
{{Main articles|Perfect group}}
Whenever a group <math>G</math> has derived subgroup equal to itself, <math>G^{(1)} =G</math>, it is called a '''perfect group'''. This includes non-abelian [[Simple group|simple groups]] and the [[Special linear group|special linear groups]] <math>\operatorname{SL}_n(k)</math> for a fixed field <math>k</math>.