Let $G$ be a discrete countable Abelian group.
We construct an infinite measure preserving rank one action $T=(T_g)$ of
$G$ such that (i) the transformation $T_g$ has infinite ergodic index but
$T_g\times T_{2g}$ is not ergodic for any element $g$ of infinite order,
(ii) $T_{g_1}\times\cdots\times T_{g_n}$ is conservative for every finite
sequence $g_1,\dots, g_n\in G$. In the case $G=\mathbb{Z}$ this answers a
question
of C. Silva. Moreover, we show that
¶ (i) every weakly
stationary nonsingular Chacon transformation with 2-cuts is power weakly
mixing and ¶ (ii) every weakly
stationary nonsingular Chacon$^*$ transformation with 2-cuts has infinite
ergodic index but is not power weakly mixing.