Are these two infinite groups isomorphic?

Question

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.

Answer 1

Every element of order $5$ in $G_2$ is itself five times another element. But this isn’t the case for $G_1$.