I have some trouble in understanding the definition of Schnorr's Identification protocol. The is the definition that I received:
Prover and verifier agree on a cyclic group $G$ of prime order $q$ generated by $g$.
The prover sends $r=g^k$ for a random $k$
The verifier sends $e \in_R Z_q$
The prover sends $s=k+xe$
The verifier now verifies if $r=g^sy^{-e}$
Now I don't understand why $r$ and $e$ have different formulas. As far as I understand, $r$ is an element of the cyclic group $G$, and $e$ also. What is the difference between:
$\in_RZ_q$
and
$g^k$ for a random $k$
mean?