One thing which I am wondering for a long time and to which I did not find an answer after doing a web search and hope to find an answer here.

When we construct the elliptic curve over a prime field why do we actually select a cyclic subgroup instead of taking the entire group of the elliptic curve?

On a side note the thing that confuses me most about this choice: We know that the cyclic subgroup of prime order p is isomorphic to Z/pZ and finding the isomorphism would mean solving the discrete log.

Switching to a cyclic group seems actually rather like making the problem easier in comparison to staying with the full elliptic curve.