  A coherent study of the production of $f_0^i$ ($i=1$, 2, 3 corresponding to
$f_0(1710)$, $f_0(1500)$, and $f_0(1370)$) in $J/\psi\to V f_0 \to V PP$ is
reported based on a previously proposed glueball and $Q\bar{Q}$ nonet mixing
scheme, and a factorization for the decay of $J/\psi\to V f_0^i$, where $V$
denotes the isoscalar vector mesons $\phi$ and $\omega$, and $P$ denotes
pseudoscalar mesons. The results show that the $J/\psi$ decays are very
sensitive to the structure of those scalar mesons, and suggest a glueball in
the $1.5-1.7$ GeV region, in line with Lattice QCD. The presence of significant
glueball mixings in the scalar wavefunctions produces peculiar patterns in the
branching ratios for $J/\psi\to V f_0^i\to VPP$, which are in good agreement
with the recently published experimental data from the BES collaboration.
