# Thread: Prove that no proper nontrivial subgroup of Z is finite.

1. ## Prove that no proper nontrivial subgroup of Z is finite.

Prove that no proper nontrivial subgroup of Z is finite. (Z is the set of integers)

2. ## Re: Prove that no proper nontrivial subgroup of Z is finite.

Hint: are the subgroups of Z cyclic?

3. ## Re: Prove that no proper nontrivial subgroup of Z is finite.

Originally Posted by dori1123
Prove that no proper nontrivial subgroup of Z is finite. (Z is the set of integers)
Another way to think about it is that if $\displaystyle G$ is some finite subgroup of $\displaystyle \mathbb{Z}$ and $\displaystyle g\in G$ then by Lagrange's theorem you know that $\displaystyle |G|g=0$. Now tell me, for what $\displaystyle g\in\mathbb{Z}$ does that equation have a solution?

4. ## Re: Prove that no proper nontrivial subgroup of Z is finite.

Another way: if $\displaystyle H\neq \{0\}$ is a subgroup of $\displaystyle \mathbb{Z}$ , use the Euclidean Division to prove that $\displaystyle H=(m)$ for some $\displaystyle m\in \mathbb{Z}-\{0\}$.

P.S. Of course, this way provides unnecessary information with respect to the initial question.