I've been reading about safe primes and their use in: Cryptography Engineering by Niels Ferguson, Bruce Schneier, and Tadayoshi Kohno.
Having a safe prime $q$ with $q=2p+1$ where $p$ is a Sophie Germain prime
They claim that the elements of the subgroup $p$ are the elements that are preferred to be used. Why?
Why not the elements in $2p$?