Is infinite product of Z a group?

Question

Having the usual coordinate-wise addition, does infinite product of $\mathbb{Z}$ forms a group?

$a,b\in \prod ^\infty \mathbb{Z}$

$a=(a_1,a_2,...)$

$b=(b_1,b_2,...)$

$a\circ b=(a_1+b_1,a_2+b_2,...)$

Then, is $(\prod ^\infty \mathbb{Z},\circ)$ a group?

Thank you very much.

Answer 1

As Pedro said, yes. It would be a good exercise to try and prove yourself that the axioms hold.