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The persistence of biological systems requires evolved mechanisms which promote stability. Cohesive primate social groups are one example of stable biological systems, which persist in spite of regular conflict. We suggest that genetic relatedness and its associated kinship structure are a potential source of stability in primate social groups as kinship structure is an important organizing principle in many animal societies. We investigated the effect of average genetic relatedness per matrilineal family on the stability of matrilineal grooming and agonistic interactions in 48 matrilines from seven captive groups of rhesus macaques. Matrilines with low average genetic relatedness show increased family-level instability such as:more sub-grouping in their matrilineal groom network, more frequent fighting with kin, and higher rates of wounding.Family-level instability in multiple matrilines within a group is further associated with group-level instability such as increased wounding. Stability appears to arise from the presence of clear matrilineal structure in the rhesus macaque group hierarchy, which is derived from cohesion among kin in their affiliative and agonistic interactions with each other. We conclude that genetic relatedness and kinship structure are an important source of group stability in animal societies,particularly when dominance and/or affilative interactions are typically governed by kinship.