: Whitehead's lemma is a technical result in

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...

used in

algebraic K-theory Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense o ...

. It states that a

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

of the form :

\begin
u & 0 \\
 0 & u^ \end

is equivalent to the

identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...

by elementary transformations (that is, transvections): :

\begin
u & 0 \\
 0 & u^ \end = e_(u^) e_(1-u) e_(-1) e_(1-u^).

Here,

e_(s)

indicates a matrix whose diagonal block is

1

and

ij^

entry is

s

. The name "Whitehead's lemma" also refers to the closely related result that the

derived group In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group (mathematics), group is the subgroup (mathematics), subgroup generating set of a group, generated by all the commutators of the group. Th ...

of the

stable general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrix, invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group (mathematics), group, because the product of two in ...

is the group generated by

elementary matrices In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GL''n''(F) when F is a field. Left multiplication (pre-multip ...

. In symbols, :

\operatorname(A) = operatorname(A),\operatorname(A) /math>.

This holds for the stable group (the

direct limit In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any categor ...

of matrices of finite size) over any ring, but not in general for the unstable groups, even over a field. For instance for :

\operatorname(2,\mathbb/2\mathbb)

one has: :

< \operatorname_2(\mathbb/2\mathbb) = \operatorname_2(\mathbb/2\mathbb) = \operatorname_2(\mathbb/2\mathbb) \cong \operatorname(3),

where Alt(3) and Sym(3) denote the alternating resp.

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...

on 3 letters.

References

Matrix theory Lemmas in linear algebra K-theory Theorems in abstract algebra {{Abstract-algebra-stub

See also

References