Check if an element is in a cyclic group

Question

As far as I understood it from this notation, if we have a cyclic group $G_q$ of order $q$, we can check if an element $a$ is included in this group, by checking if $a^q=1$.
However, I do not understand how to apply this on a practical example:
Say we have $q=7$ (and are using $g=3$ as our generator,) which would just result in the group $G_7=\{3 = 3^1 \mod 7, 2 = 3^2 \mod 7, 6, 4, 5, 1\} = \{1,2,3,4,5,6\}$. And checking if an element is included in $G_7$ can be simply done by $a<=O(G_7)=6$, but not by with $a^7 \mod 7=1$ holds.

Therefore, I would be thankful if s.o. could provide me an example of checking if an element is in an exemplary cyclic group with the method from the above-linked notation or explain to me what I got wrong here.