# Any finite subgroup of the abelian group $(F-\{0\},\cdot)$ is cyclic? ($F$ a field) [duplicate]

I found this problem:

Suppose that $F$ is a field, and that $(F-\{0\},\cdot)$ is an abelian group. Show that if $H$ is a finite subgroup of $F-\{0\}$, then $H$ is cyclic.

What I have done is:

Since $(F-\{0\},\cdot)$ is an abelian group and $H$ is finite, $H$ is a finitely generated abelian group, and by the fundamental theorem of finitely generated abelian groups, we know that $H$ is isomorphic to something like $\Bbb{Z}_{m_1} \times \cdots \times \Bbb{Z}_{m_k}$ with $m_{i}\mid m_{i+1}$. It means that $\left|H\right|= m_1 \cdots m_k$.

And I know that if I prove that $m_i$ is prime, it means that $\left|H\right|=m^k$ with $m$ prime, so it would be obvious that $H$ is cyclic. But I have no idea how to show this, so any ideas will help a lot.

Thanks.

P.S. If the problem is wrong, explain why.

## marked as duplicate by lhf, Martin Brandenburg, vadim123, Alex Becker, Henry T. HortonMay 10 '13 at 1:21

• Is $F$ given to be finite? If so, my previous answer holds, but otherwise, the answer is subtler than I'd anticipated. – Alex Wertheim May 10 '13 at 0:46