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Q/Z is divisible


Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.15

Prove that a quotient of a divisible abelian group by any proper subgroup is also divisible. Deduce that $\mathbb{Q}/\mathbb{Z}$ is divisible.


Solution: Let $A$ be a divisible abelian group and let $B < A$ be a proper subgroup. Then, let $a+B \in A/B$ and $k \in \mathbb{Z}$ with $k \neq 0$. Because $B$ is proper, $A/B$ is nontrivial, and because $A$ is abelian, $A/B$ is abelian. Since $A$ is divisible, there exists $x \in A$ such that $kx = a$; thus $k(x+B) = a+B$. Hence $A/B$ is divisible.

We saw in a previous exercise that $\mathbb{Q}$ is divisible, and $\mathbb{Z}$ is a proper subgroup of $\mathbb{Q}$. Hence $\mathbb{Q}/\mathbb{Z}$ is divisible.


Linearity

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