My brother is being slandered and defamation of character on Facebook what can he do to stop it?

One thing that he could do is file a complaint alleging libel and defamation of character against his ex-girlfriend.  To prove defamation, the plaintiff must show that the statement was (1) false and defamatory (tending to lower one's reputation in the community), (2) that the statement was "published" to a third party and was not privileged, and (3) that the publication was conducted negligently. As you can see, libel claims are complicated.  And for that reason, I suggest that you consult a lawyer in your jurisdiction specializing in defamation/libel law, media law, or internet law.  There is no guarantee that he would prevail, but if the comments are false and designed to injure his reputation, there is a chance that he could succeed.