Whitehead'S Lemma

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What is 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 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 one has: where Alt(3) and Sym(3) denote the alternating resp. symmetric group on 3 letters.

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arrangementarrayk-theorylemmalemmas in algebramathematical theoremmatricematrixmatrix theorypropositionstatementtheoremtheorems in abstract algebratheorems in algebratheory

Synonyms

Whitehead lemma

Translations

Lemma von Whitehead (de)Lemme de Whitehead (fr)

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