Whitehead's lemma

The lemma of Whitehead, named after John Henry Constantine Whitehead, is a statement from the mathematical field of ring theory. The lemma describes the commutator subgroup of the general linear group over a ring with identity.

The linear group

It is a ring with unit element. Then is the die ring, that is the amount of matrices having components, a ring with one member. This is the group of invertible elements, called the general linear group of degree. The figure

Is obviously an injective homomorphism of groups with which one can interpret as a subgroup of. The association is called linear group, sometimes stable linear group, by construction, it is the group of all invertible matrices that match up to finitely many exceptions with the infinite unit matrix.

In each group, the elementary matrices are included, they generate a subgroup and by virtue of the above homomorphism can be called a subgroup of interpret and re- form the union. Apparently is a subgroup.

Statement of the lemma by Whitehead

It is a ring with unit element. Then, that is, the commutator of. In addition, that is a perfect group.

Comments

Is a commutator subgroup is a normal subgroup in, which means you can form the factor group. This is of great importance in algebraic K-theory, where it is designated. As is the Abelisierung of, in particular, it is an abelian group.

If a body is, one is known to have a determinant figure in the group of invertible elements of the body. One can show that exactly is the core of the determinants and determinants illustration illustration therefore induces an isomorphism.

The simplest body is the residue field and by the above is a singleton and therefore. It is

Sechselementige a non- commutative group, which must therefore be isomorphic to S3. Their commutator is dreielementig, more

But is generated by the elementary matrices, that is, for the level 2 is valid. This example shows that the lemma of Whitehead does not apply to finite dimensions. So you can not give up the transition to infinite-dimensional matrices.