The term "Whitehead group" has at least two distinct definitions.
Given an associative ring 
with multiplicative
identity, 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 the elementary matrices
that add a multiple of one row to a different row. These are the matrices 
with 
and 
, where 
is the identity matrix
and 
has a single entry 1 in position 
and the zero element elsewhere.
Note that the commutativity of 
stems from the fact (Whitehead's lemma as described by
Milnor 1966) 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)
|
