Do the all points in subgroup of an elliptic curve with prime order have the same order?

Question

A subgroup $G$ of elliptic curve can constructed with point $P$ with order $q$ by $G=\langle P\rangle $.

Now, if $q$ is prime, do the all points in subgroup $G$ (except infinity point) have same order $q$?

Answer 1

Yes, of course. This is nothing to do with elliptic curves. All the elements of a group (or subgroup) have an order that divides the order of the group (or subgroup).