Final answer:
The question involves expressing algebraic structures Z140, U(140), Z400, and U(400) as products of cyclic groups using group theory principles. The decomposition is based on the prime factorization of the group's order for Zn and the structure of the group of units U(n).
Step-by-step explanation:
The main subject in question is related to the field of abstract algebra, specifically dealing with group theory. Students are asked to express certain algebraic structures, namely Z140, U(140), Z400, and U(400), as products of cyclic groups. These expressions involve finding the direct product decompositions that relate to cyclic groups, which will in turn reflect the underlying prime factorizations of the given orders of the groups.
For example, to express Z140 as a product of cyclic groups, we look at the prime factorization of 140, which is 22 × 5 × 7. This allows us to conclude that Z140 ≈ Z22 × Z5 × Z7 (isomorphic to the direct product of cyclic groups of those orders). Each part of the question requires a similar approach, albeit with the particularities of the group under consideration (whether it's the additive group of integers modulo n, Zn, or the multiplicative group of units modulo n, U(n)).