Whitehead's lemma

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For a lemma on Lie algebras, see Whitehead's lemma (Lie algebras).

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.[1][2] 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

one has:

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

See also[edit]


  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.