Title: A Group-Theoretic Framework for Knowledge Graph Embedding. Abstract: We have rigorously proved the existence of a group algebraic structure hidden in relational knowledge embedding problems, which suggests that a group-based embedding framework is essential for model design. Our theoretical analysis explores merely the intrinsic property of the embedding problem itself without introducing extra designs. Using the proposed framework, one could construct embedding models that naturally accommodate all possible local graph patterns, which are necessary for reproducing a complete graph from atomic knowledge triplets. We reconstruct many state-of-the-art models from the framework and re-interpret them as embeddings with different groups. Moreover, we also propose new instantiation models using simple continuous non-abelian groups.