Whitehead Group

There are at least two distinct notions known as the Whitehead group.

Given an associative ring A with unit, the Whitehead group associated to A is the commutative quotient group


where GL(A) is the union over all natural numbers n in N of the general linear groups GL(n,A) and where E(A) subset GL(A) is the normal subgroup generated by all elementary matrices.

Note that the commutativity of K_1A stems from the fact (proven by Whitehead) that E(A) is the commutator subgroup of GL(A).

The second definition, though different, is related to the first. Given a multiplicative group Pi with integral group ring ZPi, there exist natural homomorphisms


In this context, one can define the Whitehead group Wh(Pi) as the cokernel


See also

Abelian Group, Cokernel, Commutator Subgroup, Elementary Matrix, General Linear Group, Group Order, Group Ring, Homomorphism, Image, Multiplicative Group, Natural Number, Normal Subgroup, Quotient Group, Reduced Whitehead Group, Ring, Ring Homomorphism, Union, Unit, Unit Ring

This entry contributed by Christopher Stover

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Milnor, J. "Whitehead Torsion." Bull. Amer. Math. Soc. 72, 358-423, 1966.

Cite this as:

Stover, Christopher. "Whitehead Group." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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