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 the identity matrix by elementary transformations (that is, transvections):
Here, indicates a matrix whose diagonal block is and entry is .
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,
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
where Alt(3) and Sym(3) denote the alternating resp. symmetric group on 3 letters.
- ^ 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.
- ^ 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.