What is this abelian group notation?

Question

While studying abelian groups, I came across the abelian group $$G = \frac{1}{4} \mathbb{Z} / \mathbb{Z}$$ what is this group? I've never seen this notation before?

Answer 1

As @DavidMolano commented you can view $\frac{1}{4}\mathbb{Z}$ as a subgroup of $\mathbb{Q}$ containing $\mathbb{Z}$. Two elements $\frac{a}{4},\frac{b}{4}\in\frac{1}{4}\mathbb{Z}$ with $a,b\in\mathbb{Z}$ are congruent $\mod\mathbb{Z}$ iff $4|a-b$, and thus there is an isomorphism $\frac{1}{4}\mathbb{Z}/\mathbb{Z}\to\mathbb{Z}/4\mathbb{Z}$ defined as $\frac{a}{4}\to{}a\mod{}4$. In fact this is immediate, since every $a\in\frac{1}{4}\mathbb{Z}/\mathbb{Z}$ is congruent to $0,\frac{1}{4}, \frac{2}{4}$ or $\frac{3}{4}$.