  We present new ALMA CO(2--1) observations of two well studied group-centered
elliptical galaxies: NGC~4636 and NGC~5846. In addition, we include a revised
analysis of Cycle 0 ALMA observations of the central galaxy in the NGC~5044
group that has been previously published. We find evidence that molecular gas,
in the form of off-center orbiting clouds, is a common presence in bright
group-centered galaxies (BGG). CO line widths are $\gtrsim 10$ times broader
than Galactic molecular clouds, and using the reference Milky Way $X_{CO}$, the
total molecular mass ranges from as low as $2.6\times 10^5 M_\odot$ in NGC~4636
to $6.1\times 10^7 M_\odot$ in NGC~5044. With these parameters the virial
parameters of the molecular structures is $\gg 1$. Complementary observations
of NGC~5846 and NGC~4636 using the ALMA Compact Array (ACA) do not exhibit any
detection of a CO diffuse component at the sensitivity level achieved by
current exposures. The origin of the detected molecular features is still
uncertain, but these ALMA observations suggest that they are the end product of
the hot gas cooling process and not the result of merger events. Some of the
molecular clouds are associated with dust features as revealed by HST dust
extinction maps suggesting that these clouds formed from dust-enhanced cooling.
The global nonlinear condensation may be triggered via the chaotic turbulent
field or buoyant uplift. The large virial parameter of the molecular structures
and correlation with the warm ($10^3 - 10^5 K$)/hot ($\ge10^6$) phase velocity
dispersion provide evidence that they are unbound giant molecular associations
drifting in the turbulent field, consistently with numerical predictions of the
chaotic cold accretion process. Alternatively, the observed large CO line
widths may be generated by molecular gas flowing out from cloud surfaces due to
heating by the local hot gas atmosphere.
