Sure you can embed any group in a permutation group, but that doesn't mean the two notions behave identically in all respects. For example, two groups being iso

Sure you can embed any group in a permutation group, but that doesn't mean the two notions behave identically in all respects. For example, two groups being isomorphic is not the same as two permutation groups being isomorphic, as the latter come with their embeddings.