Whitehead group

Remarks

Example: for the trivial group, $K_1(\Z) = \set{\pm 1}$ is generated by the class of $(-1)$, so $\mathrm{Wh}(1) = 0$. Non-example: for $G = \Z/5\Z$, the group $\mathrm{Wh}(G)$ is infinite cyclic, so $T$ is not all of $K_1(\Z[G])$; there are units of $\Z[G]$ not of the form $\pm g$. $\mathrm{Wh}(G)$ is a functor of $G$, and it is abelian since $K_1$ is. One also writes $\mathrm{Wh}(G) = \tilde K_1(\Z[G]) / \set{\pm g}$ in terms of the reduced group $\tilde K_1(\Z[G]) = K_1(\Z[G])/\set{\pm 1}$.