The metal content of the hot atmospheres of galaxy groups

Galaxy groups host the majority of matter and more than half of all the galaxies in the Universe. Their hot ($10^7$ K), X-ray emitting intra-group medium (IGrM) reveals emission lines typical of many elements synthesized by stars and supernovae. Because their gravitational potentials are shallower than those of rich galaxy clusters, groups are ideal targets for studying, through X-ray observations, feedback effects, which leave important marks on their gas and metal contents. Here, we review the history and present status of the chemical abundances in the IGrM probed by X-ray spectroscopy. We discuss the limitations of our current knowledge, in particular due to uncertainties in the modeling of the Fe-L shell by plasma codes, and coverage of the volume beyond the central region. We further summarize the constraints on the abundance pattern at the group mass scale and the insight it provides to the history of chemical enrichment. Parallel to the observational efforts, we review the progress made by both cosmological hydrodynamical simulations and controlled high-resolution 3D simulations to reproduce the radial distribution of metals in the IGrM, the dependence on system mass from group to cluster scales, and the role of AGN and SN feedback in producing the observed phenomenology. Finally, we highlight future prospects in this field, where progress will be driven both by a much richer sample of X-ray emitting groups identified with eROSITA, and by a revolution in the study of X-ray spectra expected from micro-calorimeters onboard XRISM and ATHENA.