Let $G=\frac{\mathbb{Z}_{5^6}\times \mathbb{Z}_{5^4}}{\langle (15,0)\rangle}$, which is Abelian group of order $5^5$ and is isomorphic to $\mathbb{Z}_{5}\times \mathbb{Z}_{5^4}$. Suppose that the isomorphism class of $G$ is unknown to us. What I was doing there after was, to find elements of the highest order so that I can derive the isomorphism class of $G$. Here is how I proceeded.
After doing some manual computation I found that $|\alpha|=5^4$ and $|\beta|=5$ where \begin{align} &\alpha=(1,1)+\langle (15,0)\rangle\\ &\beta=(1,250)+\langle (15,0)\rangle. \end{align} and $G$ has no elements of order $5^5$.
From here it came to my mind, but could not establish it, due to which I posted here to clear my doubt.
Can we say from the above findings that, $G$ is generated by $\alpha$ and $\beta$ and hence $G$ is isomorphic to $\mathbb{Z}_{5^4}\times \mathbb{Z}_5$? If yes, then how can we establish it, and if NOT, please show what should be added to the question so that we can find the isomorphism class.
Here is the general query based on the above question: Let $G$ be a finite commutative group of order $p^n$ for some prime $p$ and $n\in \mathbb{N}$ having two elements $\alpha, \beta$ such that $|\alpha|:=p^a$ is the highest possible order and $|\beta|=p^{n-a}$. If $G$ is isomorphic to $\mathbb{Z}_{n_1}\times \mathbb{Z}_{n_2}$ then $G=\langle \alpha, \beta\rangle \simeq \mathbb{Z}_{p^a}\times \mathbb{Z}_{p^{n-a}}$