Depperm's answer, with the article/answer and Book of Mormon chapter linked therein, is much more complete than mine. However, I wish to address one part of the OP's question:
Presumably things like the Holocaust would never had happened if the transgression would not have happened.
First, let's reference 2 Nephi 2:22-23:
22 And now, behold, if Adam had not transgressed he would not have fallen, but he would have remained in the garden of Eden. And all things which were created must have remained in the same state in which they were after they were created; and they must have remained forever, and had no end.
23 And they would have had no children; wherefore they would have remained in a state of innocence, having no joy, for they knew no misery; doing no good, for they knew no sin.
Thus, in the Mormon theology, if Adam and Eve had not fallen they would have never had children. So yes, if they had not transgressed the Holocaust would not have happened, but also everything else that has happened since they transgressed would not have happened. Nobody would have been born to the earth, and Adam and Eve would have remained childless forever (or, until they chose to partake of the forbidden fruit).