# Every infinite cyclic group has non-trivial proper subgroups

1. May 4, 2017

### Mr Davis 97

1. The problem statement, all variables and given/known data
Every infinite cyclic group has non-trivial proper subgroups

2. Relevant equations

3. The attempt at a solution
I know that if we have a finite cyclic group, it only has non-trivial proper subgroups if the order of the group is not prime. But I'm not sure how to make this argument with infinite groups

2. May 4, 2017

### Staff: Mentor

What is a infinite cyclic group and how does it differ from a finite one?

3. May 4, 2017

### Mr Davis 97

Doesn't it differ by the fact that there exists no integer $n$ for which $a^n=e$, where $a$ is an element of the group besides e, the neutral element?

4. May 5, 2017

### Staff: Mentor

Yes. So we have $C = \langle a^n\,\vert \,n \in \mathbb{Z} \rangle = \{\ldots , a^{-2},a^{-1},e,a,a^2,\ldots\}$.
Non-trivial subgroup $U$ means $\{e\}\subsetneq U \subsetneq C$.
So what can we conclude from $\{e\}\subsetneq U$?
What could be a subgroup?