A thug brutally beat a woman in a Fort Greene apartment building in a terrifying, purse-snatching Christmas Day, police said yesterday.

The vicious thief followed the 28-year-old victim inside the building on Downing Street, between Fulton and Gates avenues, at around 3 a.m. and started shoving and hitting her with a piece of metal, police said.

The suspect, believed to be around 30 years old and 5-foot-10, threw the woman to the floor, stole her purse, and fled on foot, police said.