If Sophie didn't speak at the end - (Spoilers)

They would have both died if she didn't say anything

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I am working on the assumption that any children too you to survive on little food and still be able to work productively (I'm sorry to speak in such awful, inhuman terms, but that was the Nazi calculus) were gassed immediately. Auschwitz was an extermination came, and only "productive" persons survived the first few hours at a place like that.

With that in mind, it is quite possible that her Nazi tormentor was simply lying to her - both her children were going to be taken from her, but he made her suffer a hell of a lot more in the process. We don't really see what happens immediately after the girl is taken away, but if I recall correctly, she is separated from the boy in any case, and it is quite possible he was gassed. When she is working in the home of the head of the camp, she attempts to save him by begging the head to have him brought there to stay with her. Despite his promise, the boy is never brought there, and Sophie never sees or hears of him again.

My guess has always been that the Nazi tormentor was lying and the boy was killed right away as well.

Good question. Since her son was a little older and in good health I guess there was a good chance he was on his way to the children's camp (and his sister being gassed) regardless of Sophie saying anything. But as we see in the movie we don't know if died of the epidemic in the kids camp, starvation, hard labor, etc., or if he was even sent at all. So sad.