Prove that no proper nontrivial subgroup of Z is finite.

**dori1123** · July 17th 2011, 04:09 PM

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

**ModusPonens** · July 17th 2011, 05:12 PM

Hint: are the subgroups of Z cyclic?

**Drexel28** · July 17th 2011, 10:57 PM

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 $G$ is some finite subgroup of $\mathbb{Z}$ and $g\in G$ then by Lagrange's theorem you know that $|G|g=0$ . Now tell me, for what $g\in\mathbb{Z}$ does that equation have a solution?

**FernandoRevilla** · July 18th 2011, 12:11 AM

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

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

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