## Example of an Infinite Group Whose Elements Have Finite Orders

## 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.

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.

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.

Add to solve later**(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.

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.