Exercise

Suppose that H is a subgroup of G which is not necessarily normal. Which of the following statement would fail to hold concerning the map p : G -> G/H,   g -> gH

p is surjective.     p is a morphism.     p has kernel H.

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No. In the image of gH we find g.

Correct. In fact, G/H need not be a group, so it does not make sense to refer to the notion of group morphism.

No. The kernel of a map that is not a morphism has not been defined. If we take the kernel of p to be p^-1(e) (as we did for moprhisms), then it is H.