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**Hypatia** This might be a silly question, but I'm trying to figure out why the kernel of a group action is defined the way it is. If G is a group acting on a set A, the kernel of the action is defined as {g in G | ga=a for all a in A}, which is also equivalent to the intersection of all the stabilizers.

In general, the kernel of a map is the set of things that are mapped to the identity. Is there some way in which that definition matches up with the above definition? I'm sure it must, but I can't quite see how. I can see that if ga=a for all a then that is sort of the identity map, so maybe that's why, but it isn't the map that maps everything to the identity. Is there some other way of looking at a group action that makes this be the standard definition of the kernel?

Thanks.