Whitehead's lemma is a technical result in abstract algebra used in algebraic K-theory. It states that a matrix of the form
\begin{bmatrix} u&0\\ 0&u-1\end{bmatrix}
is equivalent to the identity matrix by elementary transformations (that is, transvections):
\begin{bmatrix} u&0\\ 0&u-1\end{bmatrix}=e21(u-1)e12(1-u)e21(-1)e12(1-u-1).
Here,
eij(s)
1
ij
s
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,
\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,Z/2Z)
one has:
\operatorname{Alt}(3)\cong[\operatorname{GL}2(Z/2Z),\operatorname{GL}2(Z/2Z)]<\operatorname{E}2(Z/2Z)=\operatorname{SL}2(Z/2Z)=\operatorname{GL}2(Z/2Z)\cong\operatorname{Sym}(3),