  We present IRAM 30m telescope observations of the CO(1-0) and (2-1) lines in
a sample of 11 group-dominant elliptical galaxies selected from the CLoGS
nearby groups sample. Our observations confirm the presence of molecular gas in
4 of the 11 galaxies at >4 sigma significance, and combining these with data
from the literature we find a detection rate of 43+-14%, comparable to the
detection rate for nearby radio galaxies, suggesting that group-dominant
ellipticals may be more likely to contain molecular gas than their non-central
counterparts. Those group-dominant galaxies which are detected typically
contain ~2x10^8 Msol of molecular gas, and although most have low star
formation rates (<1 Msol/yr) they have short depletion times, indicating that
the gas must be replenished on timescales ~100 Myr. Almost all of the galaxies
contain active nuclei, and we note while the data suggest that CO may be more
common in the most radio-loud galaxies, the mass of molecular gas required to
power the active nuclei through accretion is small compared to the masses
observed. We consider possible origin mechanisms for the gas, through cooling
of stellar ejecta within the galaxies, group-scale cooling flows, and gas-rich
mergers, and find probable examples of each type within our sample, confirming
that a variety of processes act to drive the build up of molecular gas in
group-dominant ellipticals.
