The group is know as "Jehovah's Witnesses". They believe that receiving blood transfusions are the same as drinking blood, which is stickily forbidden in the old testament law and repeated in the new testament book of Acts where Gentile believers were to "they keep themselves from things offered to idols, and from blood, and from strangled, and from fornication."
The problem comes down to whither or not transfusions are drinking blood or not.