  A group $G$ is called Jordan if there is a positive integer $J=J_G$ such that
every finite subgroup $\mathcal{B}$ of $G$ contains a commutative subgroup
$\mathcal{A}\subset \mathcal{B}$ such that $\mathcal{A}$ is normal in
$\mathcal{B}$ and the index $[\mathcal{B}:\mathcal{A}] \le J$ (V.L. Popov). In
this paper we deal with Jordaness properties of the groups $Bir(X)$ of
birational automorphisms of irreducible smooth projective varieties $X$ over an
algebraically closed field of characteristic zero. It is known (Yu. Prokhorov -
C. Shramov) that $Bir(X)$ is Jordan if $X$ is non-uniruled. On the other hand,
the second named author proved that $Bir(X)$ is not Jordan if $X$ is birational
to a product of the projective line and a positive-dimensional abelian variety.
  We prove that $Bir(X)$ is Jordan if (uniruled) $X$ is a conic bundle over a
non-uniruled variety $Y$ but is not birational to a product of $Y$ and the
projective line. (Such a conic bundle exists only if $\dim(Y)\ge 2$.) When $Y$
is an abelian surface, this Jordaness property result gives an answer to a
question of Prokhorov and Shramov.
