Does every infinite group has infinite cyclic subgroup?

Question

I need a help with determining if this statement is true or not.

Does every infinite group has infinite cyclic subgroup?

Answer 1

No. Just take $\mathbb{Z}_2\oplus \mathbb{Z}_2\oplus \mathbb{Z}_2 \oplus \ldots$.

Every element has finite order, so cannot generate an infinite group.

Answer 2

In the infinite abelian group $(\Bbb Q/\Bbb Z,+)$ each element has finite order, for if $0<\frac pq<1\in \Bbb Q,\ p,q\in\Bbb N$, then $q\cdot \frac pq=p=0$.

Answer 3

The quasicyclic $p$-group (the group of all radicals $\sqrt[p^n]{1}, n\in \mathbb{N}$ in $\mathbb{C}$) is infinite, but all its subgroups are finite cyclic.