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%, significantly higher than for the wider population of elliptical galaxies, and comparable to the detection rate for nearby radio galaxies. Those group-dominant galaxies which are detected typically contain ~2x10^8^M_{sun}_ of molecular gas, and although most have low star formation rates (<1M_{sun}_/yr) they have short depletion times, indicating that the gas must be replenished on timescales 10^8^yr. Almost all of the galaxies contain active nuclei, and we note that while CO detections are 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 the 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.