Consider a finite group G. For any integer $m \geq 1$ set $\gamma(m) = \gamma_G(m)$ to be the number of elements $g \in G$ such that ord($g$) = $m$. We say that $m$ is a "possible order" for G if $\gamma(m) \geq 1$, that is, if there is at least one element $g \in G$ such that ord(g) = m.
Consider the cyclic group $G = C_{36} = \{1, a, ..., a^{35} \}$. List all possible orders for G, and for each $m \geq 1$ of them calculate the value of $\gamma_G(m)$.
I understand that the order of g is the smallest integer $m$ such that $g^m = 1_G$, but how do I find out this order number? Do I go through each element in $C_{36}$ and see if I can raise it to some power ($\geq 1$) to give me the identity element?