## No Finite Abelian Group is Divisible

## Problem 240

A nontrivial abelian group $A$ is called **divisible** if for each element $a\in A$ and each nonzero integer $k$, there is an element $x \in A$ such that $x^k=a$.

(Here the group operation of $A$ is written multiplicatively. In additive notation, the equation is written as $kx=a$.) That is, $A$ is divisible if each element has a $k$-th root in $A$.

**(a)** Prove that the additive group of rational numbers $\Q$ is divisible.

**(b)** Prove that no finite abelian group is divisible.