Birational properties of linear algebraic groups

This paper investigates the question of birational equivalence of linear algebraic groups. Certain necessary conditions are derived for rationality of a group over the field of definition, which in application to algebraic tori are fairly delicate. In particular, it is shown that the condition $\textrm{Ш}(G)=0$ is a necessary criterion of rationality for tori and for semisimple groups defined over an algebraic number field.