Give an example of a nontrivial homomorphism for the givengr

Give an example of a non-trivial homomorphism for the givengroup, if an example exists. If no such homomorphism exists,explain why that is so. Z3 to Z5

Solution

no non-trivial homomorphism is possible.

suppose :Z3-->Z5 is a homomorphism.

since Z3 is generated by 1, then is completely determined by (1).

since 1 has order 3 in Z3, (0) = (1 + 1 + 1) = (1) + (1) + (1),

so (1) has either order 3 or order 1 in Z5.

but the order of (1) has to divide 5, so (1) = 0. this means is a trivial homomorphisms.