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Denote by $Z_n$ the cyclic group of order $n$. I want to determine whether two groups $G = Z_4 \times \prod_{i=1}^\infty (Z_2 \times Z_2 \times Z_4)$ and $H = (Z_2 \times Z_2) \times \prod_{i=1}^\infty (Z_2 \times Z_2 \times Z_4)$ are isomorphic or not. When observing the first factor outside of $\prod$, they seem not isomorphic. However, if I think about the map $$(x_0, (x_1^1, x_2^1, x_3^1), (x_1^2, x_2^2, x_3^2), (x_1^3, x_2^3, x_3^3), \cdots) \mapsto ((x_1^1, x_2^1), (x_1^2, x_2^2, x_0),(x_1^3, x_2^3, x_3^1), \cdots ),$$ then it looks like this map defined an isomorphism between two groups (an idea of thinking about changing the order of factors). Then why does such a problem arises at my first glance? Any comments are appreciated.

Alex Lee
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    You seem to have the intuition that for groups $A,X,Y$, if $X\times A= Y\times A$ then $X=Y$. This is true for finite groups (https://math.stackexchange.com/questions/882968/a-oplus-c-cong-b-oplus-c-is-a-cong-b-when-c-is-finite-a-and-b-infinit) but not true in general. A nice example similar to your group $G$ is to let $A=Z_2^\omega$ an infinite power of $Z_2$ and $X=Z_2$, $Y=Z_2\times Z_2$. – Frousse Apr 24 '23 at 08:03
  • Both of your groups are isomorphic to the direct product of countably many copies of Z2 and countably many copies of Z4 – Mariano Suárez-Álvarez Apr 24 '23 at 11:35

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