I have a doubt about Gallian pg 40.
An element $a^{k}$ generates a cyclic subgroup of order $d$ if and only if $\gcd(k,d)=1$.
Now let's consider the group $\mathbb Z/12\mathbb Z$ it is a cyclic group and let's take the subgroup of order $2$ which will be $\{0,6\}$. So $6$ is the generator of the subgroup and so what will be $k$ in this case? Is it $1$?
This statement appears on p85 in the only version I could find online.
There it is assumed that there is a cyclic subgroup of order $d$, generated by some element $a$, and the observation is that $a^k$ generates the same subgroup if and only if $\gcd(k,d)=1$. The crucial detail is that in context $d$ is the order of $a$, not some arbitrary number.
So if $a=1$, $\langle a\rangle$ has order $12$, and the other elements generating the same group are $a^5$, $a^7$ and $a^{11}$. If $a=6$, $\langle a\rangle$ has order $2$ and the only element generating this group is $a^1$.