There are at least two distinct notions known as the Whitehead group.
Given an associative ring with unit, the Whitehead group associated to is the commutative quotient group
(1)

where is the union over all natural numbers of the general linear groups and where is the normal subgroup generated by all elementary matrices.
Note that the commutativity of stems from the fact (proven by Whitehead) that is the commutator subgroup of .
The second definition, though different, is related to the first. Given a multiplicative group with integral group ring , there exist natural homomorphisms
(2)

In this context, one can define the Whitehead group as the cokernel
(3)
