let $G$ be a finite abelian group such that it contains a subgroup $H\_0\neq \{e\}$ which is contained in every subgroup $H\neq \{e\}$ of $G$

Prove that $G$ is cyclic.Find $o(G)$

How should I start?

**My thoughts:**
Since $G$ is a finite abelian group ,$G$ can be expressed as a direct product of cyclic groups.Let $G=\mathbb Z\_{p\_1^{\alpha\_1}}\times \mathbb Z\_{p\_2^{\alpha\_2}}\times ...\times\mathbb Z\_{p\_n^{\alpha\_n}}$.

Let $H\_0$ be a subgroup of $G$ contained in every other subgroup of $G$ .Now each $\mathbb Z\_{p\_i^{\alpha\_i}}$ can be viewed as a subgroup of $G$ .

So $H\_0$ is contained in each $\mathbb Z\_{p\_i^{\alpha\_i}}$ for each $i=1,2,...,n$. So $o(H\_0)$ divides $p\_i^{\alpha\_i}$ for each $i=1,2,...,n$ which is not possible as $p\_i$ 's are mutually prime; hence contradiction

So $i=1$ only so $G=\mathbb Z\_{p\_1^{\alpha\_1}}$ which is cyclic.Is this true??Please help.