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We study the radio and optical properties of the brightest group galaxies (BGGs) in a sample of galaxy groups from the Sloan Digital Sky Survey (SDSS) DR7. The luminosity difference between the BGG and the second-ranked galaxy in the group (known as the luminosity, or magnitude, gap) has been used as a probe for the level of galaxy interaction for the BGG within the group. We study the properties of BGGs with magnitude gaps in the range of 0–2.7 mag, in order to investigate any relation between the luminosity gap and the radio properties of the BGG. In order to eliminate selection biases, we ensure that all variations in stellar mass are accounted for. We then confirm that, at fixed stellar mass, there are no significant variations in the optical properties of the BGGs over the full range of luminosity gaps studied. We compare these optical results with the Evolution and Assembly of GaLaxies and their Environments (EAGLE) hydrodynamical simulations and find broad consistency with the observational data. Using EAGLE we also confirm that no trends begin to arise in the simulated data at luminosity gaps beyond our observational limits. Finally, we find that, at a fixed stellar mass, the fraction of BGGs that are radio-loud also shows no trend as a function of luminosity gap. We examine how the BGG offset from the center of the group may affect the radio results and find no significant trend for the fraction of radio-loud BGGs with a magnitude gap in either the BGG samples with greater or less than 100 kpc offset from the center of the group.