what is the meaning of "torsion group annihilated by integer"? Provide examples for: (1) a group that is neither torsion-free nor torsion (2) a torsion group which is not annihilated by any integer (3) a torsion-free but not free group (4) a torsion-free but not free abelian group Here is my thought and question: (1) Is $\mathbb{Z}_p\times\mathbb{Z}$ a such example? since $(1,0)$ is not torsion-free, and $(0,1)$ is not torsion. (2) What does it mean by "annihilated by any integer"? (3) Is $\mathbb{Z}^{\infty}$ an example? I am not sure. any other good example? (4) Is SL(n) an example? Thank you very much!

(1) is good: the group $\mathbb{Z}_p\times\mathbb{Z}$ is neither torsionfree because $(1,0)\
e(0,0)$ is annihilated by $p$, while $(0,1)$ has not finite order.

(2) You are required to find a torsion group $G$ such that, for every $n>0$, there exists $x\in G$ with $nx\
e0$. Hint: $\mathbb{Q}/\mathbb{Z}$

(3) You have to find a torsionfree group that is not free; is $\mathbb{Q}$ a free group?