# What is this abelian group notation?

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?

• Without knowing about the context I cannot make sure, but $\frac{1}{4}\mathbb{Z}$ is a subgroup of $\mathbb{Q}$ which contains $\mathbb{Z}$, so you can define the quotient group $\left(\frac{1}{4}\mathbb{Z}\right)/\mathbb{Z}$. There shouldn't be anything weird about the notation. – David Molano Apr 24 at 19:16

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 $$\frac{3}{4}$$.
• No, I think $\dfrac a 4 \equiv \dfrac b 4$ in $\frac14\mathbb Z/\mathbb Z$ iff $4|a-b$ (i.e., $a\equiv b \pmod 4$) – J. W. Tanner Apr 24 at 19:50