  We study large groups of birational transformations Bir(X), where X is a
variety of dimension at least 3, defined over C or a subfield of C. Two
prominent cases are when X is the projective space, in which case Bir(X) is the
Cremona group of rank n, or when X is a smooth cubic hypersurface. In both
cases, and more generally when X is birational to a conic bundle, we produce
infinitely many distinct group homomorphisms from Bir(X) to Z/2, showing in
particular that the group Bir(X) is not perfect and thus not simple. As a
consequence we also obtain that the Cremona group of rank n at least 3 is not
generated by linear and Jonqui\`eres elements.
