Find all the homomorphisms Z20 - Z6?

To find all homomorphisms from (\mathbb{Z}{20}) to (\mathbb{Z}{6}), we first note that a homomorphism is completely determined by the image of a generator of (\mathbb{Z}{20}). The generator can be taken as (1), and the image must satisfy the property that the order of the image divides the order of the domain. The order of (\mathbb{Z}{20}) is 20, and the order of (\mathbb{Z}{6}) is 6. Thus, the image of (1) can be any element in (\mathbb{Z}{6}) that, when multiplied by 20, results in 0 in (\mathbb{Z}{6}). Since (20 \equiv 2 \mod 6), the possible images are restricted to elements of order dividing 2 in (\mathbb{Z}{6}), which are (0) and (3). Therefore, the homomorphisms are given by sending (1) to (0) (the trivial homomorphism) and sending (1) to (3).