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 $\Bbb{Z}_p$.
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2See your previous question. Every group can be trivially made into a topological group by considering it with the discrete topology. – Dietrich Burde Aug 20 '19 at 13:27
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1A 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 '19 at 13:36
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@J.W.Tanner, thanks for your valuable comment and in fact it helped me a lot – MAS Aug 20 '19 at 13:43
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1Do 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 '19 at 13:46
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@reuns, thanks for details – MAS Aug 20 '19 at 13:59
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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.
Indeed, as pointed out in the comment by Dietrich Burde, any group can be trivially made into a topological group by considering it with the discrete topology.
J. W. Tanner
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1Or the trivial topology, if you do not insist on your topological groups being Hausdorff. – tomasz Aug 20 '19 at 13:48
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@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. – MAS Aug 20 '19 at 14:02