I am reading an example from Dummit and Foote's Abstract Algebra, 3rd edition.
I am wondering why the authors do not use the bar notation for the element of residue classes. For example, why do they write $\mathbb{Z}/2\mathbb{Z}=\langle~1~\rangle$ instead of $\mathbb{Z}/2\mathbb{Z}=\langle~\bar{1}~\rangle$?
The reason why I am confused is that they once use the bar notation when they introduce groups $\mathbb{Z}/n\mathbb{Z}$ and suddenly switche to this notation. Is it a common practice not to use the bar when denoting an element of $\mathbb{Z}/n\mathbb{Z}$, or is there any other reason for why they do not use the bar notation here?
As you can see the author is now discussing the subgroups of $G=Z_n$, not $\Bbb{Z}/n\Bbb{Z}$, that's why different notation is used.
In the book, the author denotes $Z_n=\{0,1,\dots,n-1\}$ and $\Bbb{Z}/n\Bbb{Z}=\{\bar{0},\bar{1},\dots,\overline{n-1}\}$