Where people cremated alive in the Holocaust?
Yes. There is at least one documented case that I’m aware of that involved German Nazis boarding up a synagogue with people still inside it and setting fire to it.
If you mean at the camps like Auschwitz - probably. I can’t imagine that the Nazis carefully inspected each and every gassed body to make sure it was completely dead before shoveling it into the fires.
I am not aware of such an incident, but it wouldn’t surprise me to hear that someone had been thrown alive and concious into the fires as well - some of the guards at those camps were really sick puppies.
In the death camps, not as a matter of course. Efficiency of the killing system always had the highest priority and the procedures were pretty rigidly adhered to. Arriving prisoners had to have clothes removed and valuables removed. Collecting such articles was an important part of the system. Having them strip “voluntarily”, gassing them, having prisoners remove the likes of gold teeth from the bodies and only then cremating them was the most efficient set of steps. Cremating anybody while still alive was just a lot more awkward.
Furthermore, the standard oven design (like these in Auschwitz) were horizontal and held about four or so bodies. You could no doubt get someone alive into one, but it’d be a messy alternative to either the standard procedure or, in the case of an individual, just shooting them.
That said, one can’t rule out it having happened on occasion. Much more likely, however, are individual prisoners being thrown alive into the mass open-air cremation pits at some of the camps. But these were again there to burn dead bodies (often exhumed) rather than as a method of killing.
What was routine was mass executing people outside the camps by rounding then up, herding them into a building (often a synagogue) and burning it down.