Question. Prove or disprove that $G_1$ and $G_2$ are isomorphic. $$G_1=\mathbb Z_5 \times \mathbb Z_{5^2}\times \mathbb Z_{5^3}\times \mathbb Z_{5^4} \times \cdots$$ $$G_2= \mathbb Z_{5^2}\times \mathbb Z_{5^3}\times \mathbb Z_{5^4} \times ...$$ My attempt: I can see that $G_1$ is isomorphic to a subgroup of $G_2$ and similarly, $G_2$ is isomorphic to a subgroup of $G_1$. But, I am unable to see how $G_1$ and $G_2$ are not isomorphic. Please help.
Assume  $$G_1=\mathbb Z_5 \times \mathbb Z_{5^2}\times \mathbb Z_{5^3}\times \mathbb Z_{5^4} \times\ldots$$ $$G_2= \mathbb Z_{5^2}\times \mathbb Z_{5^3}\times \mathbb Z_{5^4} \times \ldots$$ How do I prove $G_1$ and $G_2$ aren't isomorphic? I asked  this question here  and I received tree answer, but the answer of Ludolila will be complete when  $G_1$ and $G_2$ are not isomorphic.    Thanks in advance
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