##### Actions

An Abelian group associated with an associative ring in the following manner. It was introduced by J.H.C. Whitehead [1]. Let $A$ be an associative ring with unit element and let $\mathop{\rm GL} ( n , A )$ be the group of invertible $( n \times n )$- matrices over $A$. There are natural imbeddings

$$\mathop{\rm GL} ( 1, A) \subset \dots \subset \mathop{\rm GL} ( n , A) \subset \dots ;$$

$g \in \mathop{\rm GL} ( n, A)$ goes to

$$\left ( let  \mathop{\rm GL} ( A) = \cup _ {i=} 1 ^ \infty \mathop{\rm GL} ( i, A) . A matrix differing from the identity matrix in a single non-diagonal entry is called an [[elementary matrix]]. The subgroup  E( A) \subset \mathop{\rm GL} ( A)  generated by all elementary matrices coincides with the commutator group of  \mathop{\rm GL} ( A) . The commutator quotient group  K _ {1} A = \mathop{\rm GL} ( A) / E( A)  is called the Whitehead group of the ring  A . Let  [- 1] \in K _ {1} A  be the element corresponding to the matrix$$ \left \|

It has order 2. The quotient group $\overline{K}\; _ {1} ( A) = K _ {1} A/ \{ 0, [- 1] \}$ is called the reduced Whitehead group of the ring $A$.

Let $\Pi$ be a multiplicative group and let $\mathbf Z [ \Pi ]$ be its group ring over $\mathbf Z$. There is a natural homomorphism $j: \Pi \rightarrow \overline{K}\; _ {1} \mathbf Z [ \Pi ]$ coming from the inclusion of $\Pi \subset \mathop{\rm GL} ( 1, \mathbf Z [ \Pi ])$. The quotient group $\mathop{\rm Wh} ( \Pi ) = \overline{K}\; _ {1} \mathbf Z [ \Pi ] / j ( \Pi )$ is called the Whitehead group of the group $\Pi$.

Given a homomorphism of groups $f : \Pi _ {1} \rightarrow \Pi _ {2}$, there is a natural induced homomorphism $\mathop{\rm Wh} ( f ) : \mathop{\rm Wh} ( \Pi _ {1} ) \rightarrow \mathop{\rm Wh} ( \Pi _ {2} )$, such that $\mathop{\rm Wh} ( g \circ f ) = \mathop{\rm Wh} ( g) \circ \mathop{\rm Wh} ( f )$ for $g : \Pi _ {2} \rightarrow \Pi _ {3}$. Thus $\mathop{\rm Wh}$ is a covariant functor from the category of groups into the category of Abelian groups. If $f : \Pi \rightarrow \Pi$ is an inner automorphism, then $\mathop{\rm Wh} ( f ) = \mathop{\rm id} _ { \mathop{\rm Wh} ( \Pi ) }$.

The Whitehead group of the fundamental group of a space is independent of the choice of a base point and is essential for the definition of an important invariant of mappings, the Whitehead torsion.

#### References

 [1] J.H.C. Whitehead, "Simple homotopy types" Amer. J. Math. , 72 (1950) pp. 1–57 [2] J.W. Milnor, "Whitehead torsion" Bull. Amer. Math. Soc. , 72 (1966) pp. 358–426 [3] J.W. Milnor, "Introduction to algebraic -theory" , Princeton Univ. Press (1971)

If $A$ is commutative, the determinant and, hence, the special linear groups $\mathop{\rm SL} ( n, A)$ are defined. Using these instead of the $\mathop{\rm GL} ( n, A)$ one obtains the special Whitehead group $SK _ {1} ( A)$. One has $K _ {1} ( A) = U( A) \oplus SK _ {1} ( A)$ where $U( A)$ is the group of units of $A$.