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I have found two theorems that get at this I think: Cayley's theorem and Frucht's theorem. The former says that every group G is isomorphic to a subgroup of the symmetric group acting on G. The latter says that every finite group is the group of symmetries of a finite undirected graph.

Which one gets at the heart of what I'm asking, and also, what would one recommend to better understand groups as symmetry? My textbook takes the standard approach with defining a a group algebraically and then moving on to results, without the motivation regarding symmetry.