Let $G$ be a group and $H$ a subgroup of G. Let $G$ = $\mathbb{Z}_{10}$ and $H$ = $\left \{ 0 , 5 \right \}$.
Prove that $G$/$H$ $\cong$ $\mathbb{Z}_{5}$.
I am not exactly sure how to do this, our teacher barely explained these types of problems in lecture and I am lost. I do not even know exactly what $G$/$H$ is with respects to defined $G$ and $H$.
Use the third isomorphism theorem: $H$ is the subgroups of $\mathbf Z/10\mathbf Z$ generated by the class $5+10\mathbf Z$, i.e. $5\mathbf Z/10\mathbf Z$. So $$G/H=(\mathbf Z/10\mathbf Z)/(5\mathbf Z/10\mathbf Z)\simeq \mathbf Z/5\mathbf Z.$$