After a month of implant I have noticed pus around it and it's still swollen.My dentist told me that is normal.Is it?

It is not normal to have infection/ pus around implant it has been said that infection is " liquid bone".  Infection will cause bone loss have the site evaluated. You will need to take antibiotics to clear the infection and find out what is causing this infection ..and get the necessary treatment to prevent it from coming back. 

Sounds like infection. Take a course of Antibiotics. Assess whether implant needs to be removed. If so, graft site and replace the implant at a later date.

Pus is a sign of infection and that if left untreated can prevent proper healing of the bone and soft tissue around the implant.  In your case definitive treatment including having your dentist see you, xray the area and determine next steps including possible antibiotics and surgical intervention.