# Tagged: group of rational numbers

## Problem 594

Is it possible that each element of an infinite group has a finite order?
If so, give an example. Otherwise, prove the non-existence of such a group.

## Problem 510

Let $(\Q, +)$ be the additive group of rational numbers and let $(\Q_{ > 0}, \times)$ be the multiplicative group of positive rational numbers.

Prove that $(\Q, +)$ and $(\Q_{ > 0}, \times)$ are not isomorphic as groups.

## Problem 461

(a) Prove that the additive group $\Q=(\Q, +)$ of rational numbers is not finitely generated.

(b) Prove that the multiplicative group $\Q^*=(\Q\setminus\{0\}, \times)$ of nonzero rational numbers is not finitely generated.

## Problem 460

Let $\Q=(\Q, +)$ be the additive group of rational numbers.

(a) Prove that every finitely generated subgroup of $(\Q, +)$ is cyclic.

(b) Prove that $\Q$ and $\Q \times \Q$ are not isomorphic as groups.