- Whitehead's lemma
Whitehead's lemma is a technical result in
abstract algebra , used inalgebraic K-theory , It states that a matrix of the form:egin{bmatrix}u & 0 \ 0 & u^{-1} end{bmatrix}
is equivalent to identity by elementary transformations (here "elementary matrices" means "transvections"):
:egin{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.
It also refers to the closely related result [J. Milnor, Introduction to algebraic K -theory, Annals of Mathematics Studies 72, Princeton University Press, 1971. Section 3.1.] that the
derived group of the "stable"general linear group is the group generated byelementary matrices . 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}/2mathbb{Z}) one has::operatorname{Alt}(3) cong [operatorname{GL}_2(mathbb{Z}/2mathbb{Z}),operatorname{GL}_2(mathbb{Z}/2mathbb{Z})] < operatorname{E}_2(mathbb{Z}/2mathbb{Z}) = operatorname{SL}_2(mathbb{Z}/2mathbb{Z}) = operatorname{GL}_2(mathbb{Z}/2mathbb{Z}) cong operatorname{Sym}(3).References
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