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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

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

Here, indicates a matrix whose diagonal block is and -th 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.

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