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 so that the quotient group 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.

Contents 1 Definition 1.1 Universal property 2 Notes 3 Examples 4 Properties 5 See also

[edit] Definition

Given a group G the commutator subgroup [G,G] (also denoted G′ or G⁽¹⁾) of G is the subgroup generated by all the commutators of elements of G, that is

This construction can be iterated:

G⁽⁰⁾: = G

A group with G⁽ⁿ⁾ = {e} for some n in N is called a solvable group.

The quotient group G / [G,G] is an abelian group called the abelianization of G or G made abelian. It is usually denoted by G^ab. The abelianization of G coincides with the first homology group of G.

A group G is called a perfect group if [G,G] = G. Thus the abelianization of a perfect group is trivial.

[edit] Universal property

The commutator subgroup satisfies the following universal property:

Given a group G, the commutator subgroup [G,G] is the uniquely defined (up to isomorphism) subgroup of G so that given any homomorphism f : G → A from G to an abelian group A and the projection π : G → G/[G,G] then there exists a unique homomorphism s : G/[G,G] → A such that s o π = f

In other words, G^ab=G/[G,G] is the maximal abelian quotient of G.

In the language of category theory the functor which assigns to every group its abelianization is left adjoint to the forgetful functor which assigns to every abelian group its underlying group.

[edit] 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 example is a group of size 96.

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.

[edit] Examples

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

[edit] 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 f([G,G]) is a subgroup of [H,H], 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 [G,G] 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[G,G]g = [G,G] for every g in G, and therefore that [G,G] is a normal subgroup of G.).

[edit] See also

• solvable group
• nilpotent group