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.
Add to solve laterLet $\Q=(\Q, +)$ 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.
Add to solve later