# Abelian groups of finite order

Show that an abelian group of order 75 has a cyclic subgroup of order 15.

Do I need to use the fundamental theorem of finite abelian groups in some way?

• Yes, you can use the fundamental theorem of finite ablian groups. What do you get? Feb 26 '14 at 16:15
• "Can" is not same as "need to": no @Nicholas, you do not need to use that theorem...but you can, of course. Feb 26 '14 at 17:31

Yes, you can certainly use the Fundamental Theorem of finitely generated abelian groups to list all possible non-isomorphic abelian groups of order $75$.
To help distinguish between those abelian groups that are isomorphic vs. those that are not isomorphic, use the fact that $$\mathbb Z_{mn} \cong \mathbb Z_m\times \mathbb Z_n \iff \gcd(m, n) = 1$$
Then show that each of them necessarily has a subgroup of order $15$.
By Cauchy's theorem, there is an element $u$ of order $3$ and an element $v$ of order $5$. Since $3$ and $5$ are coprime and the group is abelian, $uv$ has order $3\cdot 5=15$.