Question

(i) Prove that Z/⟨60⟩ is isomorphic to Z/⟨3⟩ × Z/⟨4⟩ × Z/⟨5⟩. (ii) Exhibit elements 1, 2, and 3 in Z/⟨60⟩ such that ₁² = ₁, ₂² = ₂, and ₃² = ₃, and ₁ ₂ = ₂ ₁, ₂ ₃ = ₃ ₂, and ₁ ₃ = ₃ ₁, satisfying the properties of an isomorphism from Z/⟨60⟩ to Z/⟨3⟩ × Z/⟨4⟩ × Z/⟨5⟩.

Answer 1

To prove the isomorphism between Z/⟨60⟩ and Z/⟨3⟩ × Z/⟨4⟩ × Z/⟨5⟩, we can define a map f: Z/⟨60⟩ → Z/⟨3⟩ × Z/⟨4⟩ × Z/⟨5⟩. This map can be shown to be a well-defined homomorphism, demonstrating the isomorphism between the two groups.

Step-by-step explanation:

To prove that Z/⟨60⟩ is isomorphic to Z/⟨3⟩ × Z/⟨4⟩ × Z/⟨5⟩, we need to show that there exists an isomorphism between the two groups.

Let's define the map f: Z/⟨60⟩ → Z/⟨3⟩ × Z/⟨4⟩ × Z/⟨5⟩ as follows: f([a]60) = ([a]3, [a]4, [a]5), where [a]60 is the equivalence class of a modulo 60.

We can show that f is a well-defined homomorphism that is both injective and surjective, which proves the isomorphism between the two groups