Galaxy groups and clusters are believed to influence galaxy evolution. It has been shown that the groups with early formation epoch, also known as fossil groups, differ in halo and IGM properties, compared to the general population of galaxy groups. However, there is a controversy over the properties of the brightest group galaxies which may have been affected by the group's dynamical state. I will focus on two properties of the brightest group galaxies, the AGN activity and the stellar population. The groups with early formation epoch, or dynamically old, host under luminous AGNs in radio relative to those hosted by dynamically young groups. There is no evidence that such a distinction exists in the stellar population of these galaxies, leaving the debate open whether this is an observational limitation.