What does $\langle\overline{4}\rangle$ mean?

Question

What does $\langle\overline{4}\rangle$ mean?

Context: Find the right cosets of $H<G$ and $[G:H]$ where $G=\mathbb{Z}_{12}$ and $H=\langle\overline{4}\rangle$

Answer 1

Here $\bar{4}=\{4+12k\mid k\in\mathbb{Z}\}$ is an element of $\Bbb Z_{12}$ and so $\langle \bar{4}\rangle$ is the subgroup of $\Bbb Z_{12}$ generated by $\bar{4}$; that is, the group of all powers of $\bar{4}$ in $\Bbb Z_{12}$, which, since the operation is additive, are all multiples of $\bar{4}$.