What does Elizabeth ask John to do? question is in second act of story
Elizabeth asks John to go to Abigail and prove to her that their relationship is truly over.
Here is the scene, from Act 2:
Elizabeth: ...Would you favor me with this? Go to Abigail.
Proctor (his soul hardening as he senses): What have I to say to Abigail?
Elizabeth (delicately): John-- grant me this. You have a faulty understanding of young girls. There is a promise made in any bed --
Proctor: (Striving against his anger): What promise!
Elizabeth: Spoke or silent, a promise is surely made. And she may dot on it now -- I am sure she does -- and thinks to kill me, then to take my place.
Proctor: When will you know me, woman? Were I stone I would have cracked for shame this seven month!
Elizabeth: Then go to her and tell her she's a whore. Whatever promise she may sense -- break it, John, break it.
Elizabeth asks John to go to the court and reveal what he has learned from Abigail. Abigail has told John that the girls were only dancing in the woods and that it wasn't a witchcraft incident. Elizabeth is upset with John because she believes that his reluctance to inform the court is based on his lingering love for Abigail. John, however, claims that his reluctance to inform the court is based on his lack of proof. When Abigail gave him the news about dancing, the two were alone. This, by the way, further propels Elizabeth's distrust of John, as she was unaware that Abigail and John were alone.