# Is there a standard name for this infinite group?

Consider the group of sequences $$\{(a_1,a_2,\dots): a_i\in\mathbb{Z}/2\mathbb{Z}\}$$ where the group operation is component-wise addition. Is there a standard name for this group, such as $(\mathbb{Z}/2\mathbb{Z})^{\infty}$, $(\mathbb{Z}/2\mathbb{Z})^{\mathbb{N}}$, or something similar? It is isomorphic to $\mathbb{Z}/2\mathbb{Z}[x]$ under addition, but I want to emphasize the additive group structure and not assign it any multiplicative structure.

EDIT: Silly me, it's not isomorphic to $\mathbb{Z}/2\mathbb{Z}[x]$. See below.

• I would call it $\prod_{\mathbb Z} \mathbf C_2$ or $\prod_{\mathbb Z} \mathbf (\mathbb Z/2\mathbb Z)$. But it is not isomorphic to $\mathbb Z/2\mathbb Z[x]$ because there only a finite number of terms can be non-zero, this latter group I would denote $\bigoplus_{\mathbb Z} \mathbf C_2$. Commented Nov 27, 2015 at 23:44

It would be standard to call this group $(\mathbb{Z}/2\mathbb{Z})^\mathbb{N}$ or $(\mathbb{Z}/2\mathbb{Z})^\omega$, or perhaps $\prod_\mathbb{N}\mathbb{Z}/2\mathbb{Z}$. It has probably also been called $(\mathbb{Z}/2\mathbb{Z})^\infty$, but I would recommend avoiding that notation, as it often (possibly more often) refers to the subgroup of your group consisting of sequences which have only finitely many nonzero entries. Incidentally, $\mathbb{Z}/2\mathbb{Z}[x]$ is not isomorphic to your group, rather it is isomorphic to this subgroup of sequences with finite support. The full sequence space is isomorphic instead to (the underlying additive group of) the power series ring $\mathbb{Z}/2\mathbb{Z}[[x]]$.
• Or, it is equal to $\mathrm(Hom)(\mathbb F_2[x],\mathbb F_2)$ where the homomorphisms are group homomorphism. They are dual vectors spaces over $\mathbb F_2$. Commented Nov 27, 2015 at 23:53