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)
|
