# What is the topology on the group $G =\prod_{n=1}^{\infty} \mathbb{Z}/(p^n)$ can be given?

What is the topology on the group $$G =\prod_{n=1}^{\infty} \mathbb{Z}/(p^n)$$ can be given so that it becomes a topological group ?

Here $$\mathbb{Z}/(p^n)$$ is finite cyclic group.

Is it the $$I-$$adic topology for ideal $$I$$ or is it the Zarisky topology?

But in that case we need ring structure.

I am confused. Is there product topology?

• Yes, see here. – Dietrich Burde Aug 19 at 14:04
• The default topology would be the product topology of the discrete topologies. The result will not be discrete. – Thomas Andrews Aug 19 at 14:08
• There's no "the" topology. There are infinitely many different topologies on your $G$ that turn it into a topological group. What exactly are you trying to do here? What's your context? – freakish Aug 19 at 14:21
• Each group $\mathbb{Z}/p^n$ is discrete. I did write this above. – Tyrone Aug 20 at 9:15
• @BijanDatta, A non-trival group $G$ can always be endowed with various distinct topologies making it a topological group. Two "universal choices" are the discrete topology and the indiscrete topology. Since $\mathbb{Z}/p^n$ is a group it also can be endowed with discrete metric – M. A. SARKAR Aug 20 at 11:03