Whitehead's lemma

Whitehead's lemma is a technical result in abstract algebra, used in algebraic K-theory, It states that a matrix of the form

:

is equivalent to identity by elementary transformations (here "elementary matrices" means "transvections"):

:

Here, $e\_\{ij\}(s)$ indicates a matrix whose diagonal block is $1$ and $ij^\{th\}$ entry is $s$.

It also refers to the closely related result [J. Milnor, Introduction to algebraic K -theory, Annals of Mathematics Studies 72, Princeton University Press, 1971. Section 3.1.] that the derived group of the "stable" general linear group is the group generated by elementary matrices. In symbols, $operatorname\{E\}(A)\; =\; [operatorname\{GL\}(A),operatorname\{GL\}(A)]$.

This holds for the stable group (the direct limit of matrices of finite size) over any ring, but not in general for the unstable groups, even over a field. For instance for $operatorname\{GL\}(2,mathbb\{Z\}/2mathbb\{Z\})$ one has::

References