Let me add to this by talking about cyclic groups. All cyclic groups are equivalent to addition modulo an integer, but they can be different computationally: to go from an element of your cyclic group to the equivalent integer may be very difficult. For addition modulo n, the computational Diffie-Hellman problem is effectively:
Given integers 1, a, and b, compute ab. This is obviously easy. The problem of translating from an arbitrary cyclic group to addition mod n is the discrete log problem, so perhaps that explains why it is important to cryptography that it be hard for your group.
Given integers 1, a, and b, compute ab. This is obviously easy. The problem of translating from an arbitrary cyclic group to addition mod n is the discrete log problem, so perhaps that explains why it is important to cryptography that it be hard for your group.