In addition to Jews and other groups, homosexuals were imprisoned and put to death (despite many top Nazis being homosexuals). So, I guess the right has something to cheer about.
Hitler certainly gave the order to kill the Jews. Adolf Eichmann, who was in charge of the operation, later said so, at any rate. He tried to do his best to carry out the orders. The order for mass death of all Jews was in writing, and still exist. The Nazis were, if anything, excellent record keepers.
As with most conspiracies theories, the holocaust deniers weave thin theory out of scant and misleading "evidence," which is easily buried by massive amounts of real-world proof. That which hints at proof of the theory is clung to long after it's been debunked, while any contravening proof is dismissed via tortured logic.