# How the finite cyclic group $\Bbb{Z}_p$ can be endowed with discrete topology to make it a topological group?

How the finite cyclic group $$\Bbb{Z}_p$$ can be endowed with discrete topology to make it a topological group?

We have information that in discrete topology all subsets of $$\Bbb{Z}_p$$ is open set and it is the largest topology on $$\Bbb{Z}_p$$.

• – Dietrich Burde Aug 20 at 13:27
• A topological group is a group with a topology where the group operation and inverse are continuous; $\Bbb Z_p$ is a group, the discrete topology is a topology, and any function from a discrete topological space is continuous – J. W. Tanner Aug 20 at 13:36
• @J.W.Tanner, thanks for your valuable comment and in fact it helped me a lot – M. A. SARKAR Aug 20 at 13:43
• Do you see that $d(x,y) = 1$ if $x\ne y$, $d(x,y)=0$ otherwise is a metric on $G$ ? This makes $G$ into a topological group. This metric is the discrete metric. Any function from $G \to \Bbb{C}$ is continuous and the Haar measure (ie. the $G$-invariant integral) is $\int_G f(x)dx = \sum_{x \in G} f(x)$ – reuns Aug 20 at 13:46
• @reuns, thanks for details – M. A. SARKAR Aug 20 at 13:59

$$\Bbb Z_p$$ is a group, the discrete topology is a topology, and any function from a discrete topological space is continuous.
• @J.W. Tanner, the inverse map $f: \Bbb{Z}_p \to \Bbb{Z}_p$ is continuous but how to see that the map $g:\Bbb{Z}_p \times \Bbb{Z}_p \to \Bbb{Z}_p$ is continuous ? Because in this case $\Bbb{Z}_p \times \Bbb{Z}_p$ is not discrete topological space. – M. A. SARKAR Aug 20 at 14:02