Whitehead's lemma

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

$\begin{bmatrix} u & 0 \\ 0 & u^{-1} \end{bmatrix}$

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

$\begin{bmatrix} u & 0 \\ 0 & u^{-1} \end{bmatrix} = e_{21}(u^{-1}) e_{12}(1-u) e_{21}(-1) e_{12}(1-u^{-1}).$

Here, $e i j (s)$ indicates a matrix whose diagonal block is $1$ and $i j t h$ entry is $s$ .

It also refers to the closely related result^[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 $\operatorname{GL}(2,\mathbb{Z}/2\mathbb{Z})$ one has:

$\operatorname{Alt}(3) \cong [\operatorname{GL}_2(\mathbb{Z}/2\mathbb{Z}),\operatorname{GL}_2(\mathbb{Z}/2\mathbb{Z})] < \operatorname{E}_2(\mathbb{Z}/2\mathbb{Z}) = \operatorname{SL}_2(\mathbb{Z}/2\mathbb{Z}) = \operatorname{GL}_2(\mathbb{Z}/2\mathbb{Z}) \cong \operatorname{Sym}(3).$