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

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

is equivalent to the identity matrix by elementary transformations (that is, 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, $e_{{ij}}(s)$ 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.^[1][2] 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:

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

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

1. Milnor, John Willard (1971). Introduction to algebraic K-theory. Annals of Mathematics Studies. 72. Princeton, NJ: Princeton University Press. Section 3.1. MR 0349811. Zbl 0237.18005.
2. Snaith, V. P. (1994). Explicit Brauer Induction: With Applications to Algebra and Number Theory. Cambridge Studies in Advanced Mathematics. 40. Cambridge University Press. p. 164. ISBN 0-521-46015-8. Zbl 0991.20005.