# Are they isomorphic?

$G$ and $G \times G$ where $G = \Bbb Z_2 \times \Bbb Z_2 \times \Bbb Z_2 \times\cdots$

The answer says yes but I cannot figure out what homomorphism function I could use.

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@Juan Liner: Okay, that was a bit snarky. Here's one way to think about it and so slap your forehead about how obvious it all was: Think of $G$ as being a product indexed by the natural numbers, one copy for every natural number. That is the same as having $G$ be a product indexed by the even natural numbers; or as having $G$ indexed by the odd natural numbers. Just the same, right? Well, now, when you look at $G\times G$, think of the first $G$ as being indexed by the even natural numbers, and the second as being indexed by the odd natural numbers. Then $G\times G$ is indexed by... – Arturo Magidin Nov 11 '10 at 5:24

Think of $$G = \mathbb{Z_{2_1}}\times \mathbb{Z_{2_2}} \times \mathbb{Z_{2_3}} \times \mathbb{Z_{2_4}} \times \mathbb{Z_{2_5}} \times \ldots$$ and $$G \times G= (\mathbb{Z_{2_1}}\times \mathbb{Z_{2_3}} \times \mathbb{Z_{2_5}} \times \ldots) \times (\mathbb{Z_{2_2}}\times \mathbb{Z_{2_4}} \times \mathbb{Z_{2_6}} \times \ldots)$$