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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

{\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}}=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 $s$ .

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,\mathbb {Z} /2\mathbb {Z} )

one has:

\operatorname {Alt} (3)\cong [\operatorname {GL} _{2}(\mathbb {Z} /2\mathbb {Z} ),\operatorname {GL} _{2}(\mathbb {Z} /2\mathbb {Z} )]<\operatorname {E} _{2}(\mathbb {Z} /2\mathbb {Z} )=\operatorname {SL} _{2}(\mathbb {Z} /2\mathbb {Z} )=\operatorname {GL} _{2}(\mathbb {Z} /2\mathbb {Z} )\cong \operatorname {Sym} (3),

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

References

^ Milnor, John Willard (1971). Introduction to algebraic K-theory. Annals of Mathematics Studies. Vol. 72. Princeton, NJ: Princeton University Press. Section 3.1. MR 0349811. Zbl 0237.18005.
^ Snaith, V. P. (1994). Explicit Brauer Induction: With Applications to Algebra and Number Theory. Cambridge Studies in Advanced Mathematics. Vol. 40. Cambridge University Press. p. 164. ISBN 0-521-46015-8. Zbl 0991.20005.

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[Mil31-1] Milnor, John Willard (1971). Introduction to algebraic K-theory. Annals of Mathematics Studies. Vol. 72. Princeton, NJ: Princeton University Press. Section 3.1. MR 0349811. Zbl 0237.18005.

[Sn164-2] Snaith, V. P. (1994). Explicit Brauer Induction: With Applications to Algebra and Number Theory. Cambridge Studies in Advanced Mathematics. Vol. 40. Cambridge University Press. p. 164. ISBN 0-521-46015-8. Zbl 0991.20005.

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[2]

@@ Line 1: / Line 1: @@
+:{{For|a lemma on Lie algebras|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 (mathematics)|matrix]] of the form
+'''Whitehead's lemma''' is a technical result in [[abstract algebra]] used in [[algebraic K-theory]].  It states that a [[matrix (mathematics)|matrix]] of the form
-:<math>
+:<math>\begin{bmatrix}
-\begin{bmatrix}
 u & 0 \\
-& u^{-1} \end{bmatrix}</math>
+& u^{-1}
+\end{bmatrix}</math>
-is equivalent to [[identity matrix|identity]] by [[elementary matrices|elementary transformations]] (here "elementary matrices" means "transvections"):
+is equivalent to the [[identity matrix]] by [[elementary matrices|elementary transformations]] (that is, transvections):
-:<math>
+:<math>\begin{bmatrix}
-\begin{bmatrix}
 u & 0 \\
-& u^{-1} \end{bmatrix} = e_{21}(u^{-1}) e_{12}(1-u) e_{21}(-1) e_{12}(1-u^{-1}). </math>
+& u^{-1} \end{bmatrix} = e_{21}(u^{-1}) e_{12}(1-u) e_{21}(-1) e_{12}(1-u^{-1}).</math>
-Here, <math>e_{ij}(s)</math> indicates a matrix whose diagonal block is <math>1</math> and <math>ij^{th}</math> entry is <math>s</math>.
+Here, <math>e_{ij}(s)</math> indicates a matrix whose diagonal block is <math>1</math> and <math>ij</math>-th entry is <math>s</math>.
+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 (mathematics)|group]] generated by [[elementary matrices]].<ref name=Mil31>{{cite book | last1=Milnor | first1=John Willard | author1-link= John Milnor | title=Introduction to algebraic K-theory | publisher=[[Princeton University Press]] | location=Princeton, NJ | mr=0349811 | year=1971 | zbl=0237.18005 | series=Annals of Mathematics Studies | volume=72 | at=Section 3.1 }}</ref><ref name=Sn164>{{cite book | title=Explicit Brauer Induction: With Applications to Algebra and Number Theory | volume=40 | series=Cambridge Studies in Advanced Mathematics | first=V. P. | last=Snaith | publisher=[[Cambridge University Press]] | year=1994 | isbn=0-521-46015-8 | zbl=0991.20005 | page=[https://archive.org/details/explicitbrauerin0000snai/page/164 164] | url=https://archive.org/details/explicitbrauerin0000snai/page/164 }}</ref> In symbols,
-It also refers to the closely related result<ref>[[J. Milnor]], Introduction to algebraic K -theory, Annals of Mathematics Studies 72, Princeton University Press, 1971. Section 3.1.</ref> that the [[derived group]] of the ''stable'' [[general linear group]] is the group generated by [[elementary matrices]]. In symbols, <math>\operatorname{E}(A) = [\operatorname{GL}(A),\operatorname{GL}(A)]</math>.
+:<math>\operatorname{E}(A) = [\operatorname{GL}(A),\operatorname{GL}(A)]</math>.
-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 <math>\operatorname{GL}(2,\mathbb{Z}/2\mathbb{Z})</math> one has:
+This holds for the stable group (the [[direct limit of groups|direct limit]] of matrices of finite size) over any [[ring (mathematics)|ring]], but not in general for the unstable groups, even over a [[field (mathematics)|field]]. For instance for
+:<math>\operatorname{GL}(2, \mathbb{Z}/2\mathbb{Z})</math>
+one has:
-:<math>\operatorname{Alt}(3) \cong [\operatorname{GL}_2(\mathbb{Z}/2\mathbb{Z}),\operatorname{GL}_2(\mathbb{Z}/2\mathbb{Z})] < \operatorname{E}_2(\mathbb{Z}/2\mathbb{Z}) = \operatorname{SL}_2(\mathbb{Z}/2\mathbb{Z}) = \operatorname{GL}_2(\mathbb{Z}/2\mathbb{Z}) \cong \operatorname{Sym}(3).</math>
+:<math>\operatorname{Alt}(3) \cong [\operatorname{GL}_2(\mathbb{Z}/2\mathbb{Z}),\operatorname{GL}_2(\mathbb{Z}/2\mathbb{Z})] < \operatorname{E}_2(\mathbb{Z}/2\mathbb{Z}) = \operatorname{SL}_2(\mathbb{Z}/2\mathbb{Z}) = \operatorname{GL}_2(\mathbb{Z}/2\mathbb{Z}) \cong \operatorname{Sym}(3),</math>
+where Alt(3) and Sym(3) denote the [[alternating group|alternating]] resp. [[symmetric group]]<!--- I suppose this is meant; that article does not mention "Sym(n)" notation---> on 3 letters.
 ==See also==
-*[[Special linear group#Relations to other subgroups of GL(n,A)]]
+*[[Special linear group#Relations to other subgroups of GL(n, A)|Special linear group#Relations to other subgroups of GL(''n'', ''A'')]]
 ==References==
 <references/>
+[[Category:Matrix theory]]
-{{algebra-stub}}
-[[Category:Linear algebra]]
+[[Category:Lemmas in linear algebra]]
-[[Category:Lemmas]]
 [[Category:K-theory]]
+[[Category:Theorems in abstract algebra]]
+{{matrix-stub}}