Commutator subgroup

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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. The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian. So in some sense it provides a measure of how far the group is from being abelian; the larger the commutator subgroup is, the "less abelian" the group is.

1 Definition
2 Notes
3 Examples
4 Properties
5 Footnotes
6 See also

Definition

Given a group G the commutator subgroup [G,G] (also called the derived subgroup, and denoted G′ or G⁽¹⁾) of G is the subgroup generated by all the commutators^[1] <math>[g,h]:=ghg^{-1}h^{-1}</math> of elements of G, that is

<math>[G,G] = \langle ghg^{-1}h^{-1} \, | \, g, h \in G\rangle .</math>

The commutator subgroup is a fully characteristic subgroup: it is closed under all endomorphisms of the group, which is stronger than normal.

Derived Series

This construction can be iterated:

<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. 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, and one can continue it to infinite ordinal numbers via transfinite induction.

Special Classes of Groups

A group G is an abelian group if and only if the derived group is trivial: <math>[G,G]=e</math>. Equivalently, if and only if the group equals its abelianization. A group G is a perfect group if and only if the derived group equals the group itself: <math>[G,G]=G</math>. Equivalently, if and only if the abelianization of the group is trivial. This is "opposite" to abelian. A group with G⁽ⁿ⁾ = {e} for some n in N is called a solvable group; this is weaker than abelian.

Abelianization

The quotient group <math>G/[G,G]</math> is an abelian group called the abelianization of G or G made abelian. It is the "largest" abelian group to which G maps, in the sense of a universal property. It is usually denoted by G^ab. The abelianization of G coincides with the first homology group of G.

Lower Central Series

An alternate construction can be iterated:

<math>G'^{(n)} := [G'^{(n-1)},G] \quad n \in \mathbf{N}</math>

This is called the lower central series, and a group with <math>G'^{(n)} = \{e\}</math> for some n in N is called a nilpotent group; this is stronger than being solvable.

Notes

In general the set of all commutators of the group is not a subgroup so we have to consider the subgroup generated by them. The smallest examples are two non-isomorphic groups of order 96. In each of these examples, the elements of the derived subgroup may be written as a product of two commutators. The commutator subgroup can also be defined as the set of elements g of the group which have an expression as a product g=g₁g₂...g_k that can be rearranged to give the identity.

Examples

The commutator subgroup of the alternating group A₄ is the Klein four group
The commutator subgroup of the symmetric group <math>S_n</math> is the alternating group <math>A_n</math>
The commutator subgroup of the quaternion group Q = {1, −1, i, −i, j, −j, k, −k} is [Q,Q]={1, −1}.

Properties

A group is abelian if and only if its commutator subgroup is the trivial group {e}. Given a group G, a factor group G/N is abelian if and only if [G,G] ⊂ N. If f : G → H is a group homomorphism, then <math>f([G,G])</math> is a subgroup of <math>[H,H]</math>, because f maps commutators to commutators. This implies that the operation of forming derived groups is a functor from the category of groups to the category of groups. Applying this to endomorphisms of G, we find that <math>[G,G]</math> is a fully characteristic subgroup of G, and in particular a normal subgroup of G. (To reach the final conclusion, simply take conjugation with any particular g in G to be the automorphism in question. We see that g^-1<math>[G,G]</math>g = <math>[G,G]</math> for every g in G, and therefore that <math>[G,G]</math> is a normal subgroup of G. This is shown explicitly below).

Category theory

In the language of category theory, abelian groups are a reflective subcategory of the category of groups, and abelianization is the reflector: the functor which assigns to every group its abelianization is left adjoint to the inclusion of abelian groups in groups. In other words, G^ab=G/[G,G] is the maximal abelian quotient of G. In terms of universal properties, the commutator subgroup satisfies the following:

Given a group G, the commutator subgroup [G,G] is the uniquely defined subgroup^[2] of G so that given any homomorphism <math>f\colon G \to A</math> from G to an abelian group A, then there exists a unique homomorphism <math>s\colon G/[G,G] \to A</math> such that <math>s \circ \pi = f</math>, where <math>\pi</math> is the abelianization <math>\pi\colon G \to G/[G,G]</math> .

Normality Property

The commutator subgroup is a normal. That is, <math>[G,G]\triangleleft G</math> Proof: If g in G and h in [G,G] then <math>h^g = h [h,g]</math> is the product of two elements of [G,G]. Furthermore, if H is a subgroup of G that contains [G,G], then H is normal in G. Proof: If g in G and h in H, then <math>h^g = h [h,g]</math> is the product of two elements of H.

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

Footnotes

^ Some authors use the opposite convention of <math>[g,h]:=g^{-1}h^{-1}gh</math>; this does not change the definition of commutator subgroup, as inverting switches between conventions: <math>[g^{-1},h^{-1}]=g^{-1}h^{-1}gh</math>.
^ <math>[G,G]</math> is a well-defined subgroup, since it is (fully) characteristic; it is not only "defined up to unique isomorphism", as is common in universal properties, as any isomorphism of G stabilizes it.