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

$\left[\begin{array}{cc}u& 0\\ 0& u^{-1}\end{array}\right]$

is equivalent to the identity matrix by elementary transformations (that is, transvections):

$\left[\begin{array}{cc}u& 0\\ 0& u^{-1}\end{array}\right]=e_{21}(u^{-1})e_{12}(1-u)e_{21}(-1)e_{12}(1-u^{-1}).$

Here, $e_{ij}(s)$ indicates a matrix whose diagonal block is $1$ and $ij$-th entry is $s$.

The name "Whitehead's lemma" also refers to the closely related result that the derived group of the stable general linear group is the group generated by elementary matrices. In symbols,

$\mathrm{E}(A)=[\mathrm{GL}(A),\mathrm{GL}(A)]$.

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

$\mathrm{GL}(2,\mathbb{Z}/2\mathbb{Z})$

one has:

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

where Alt(3) and Sym(3) denote the alternating resp. symmetric group on 3 letters.

See also

• Special linear group#Relations to other subgroups of GL(n, A)

References

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