Using Chandra data for a sample of 26 galaxy groups, we constrained the central cooling times (CCTs) of the ICM and classified the groups as strong cool-core (SCC), weak cool-core (WCC) and non-cool-core (NCC) based on their CCTs. The total radio luminosity of the brightest cluster galaxy (BCG) was obtained using radio catalog data and literature, which was compared to the CCT to understand the link between gas cooling and radio output. We determined K-band luminosities of the BCG with 2MASS data, and used it to constrain the masses of the SMBH, which were then compared to the radio output. We also tested for correlations between the BCG luminosity and the overall X-ray luminosity and mass of the group. The observed cool-core/non-cool-core fractions for groups are comparable to those of clusters. However, notable differences are seen. For clusters, all SCCs have a central temperature drop, but for groups, this is not the case as some SCCs have centrally rising temperature profiles. While for the cluster sample, all SCC clusters have a central radio source as opposed to only 45% of the NCCs, for the group sample, all NCC groups have a central radio source as opposed to 77% of the SCC groups. For clusters, there are indications of an anticorrelation trend between radio luminosity and CCT which is absent for the groups. Indications of a trend of radio luminosity with black hole mass observed in SCC clusters is absent for groups. The strong correlation observed between the BCG luminosity and the cluster X-ray luminosity/cluster mass weakens significantly for groups. We conclude that there are important differences between clusters and groups within the ICM cooling/AGN feedback paradigm.
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