ИСТИНА

Let G be a connected algebraic group defined over the field of real numbers R. The set of complex points G(C) is a connected complex Lie group, while the set of real points G(R) is a real Lie group, which is not necessarily connected: consider, e.g., GL_n(R) as a counterexample. It turns out that the group of connected components of G(R) is always an elementary finite Abelian 2-group. This result was obtained first by H. Matsumoto in 1964 for semisimple algebraic groups. Generalizing and specifying Matsumoto's theorem, we compute the component group of G(R) explicitly for an arbitrary (not necessarily linear) connected algebraic group. The computation makes use of Galois cohomology over real numbers. Our result looks most transparent in the cases where G is a linear algebraic group or an Abelian variety.