Count the number of group homomorphisms from a group of order 24 to a group of order 25.
This question appeared on a past paper at my University directly following a proof that if $f : G \to H $ is an injective group homomorphism, then $ \forall g \in G \quad |g| = |f(g)| $.
I can see that this means that a group homomorphism from a group of order 24 to a group of order 25 cannot be injective, but I don't know how to approach the case in which it isn't.
The image of the homomorphism is a group of order 1, 5 or 25 by Lagrange's theorem. It cannot be 25, since the domain is only of order 24; it cannot be 5, since there would then be elements of order 5 in the image, implying that there is an element of order a multiple of 5 in the domain, but this cannot happen because 5 does not divide 24, again by Lagrange's theorem.
Thus the image is trivial and there is exactly one homomorphism between the two groups.