Final answer:
The isomorphism type of an Abelian group of order 600 is (c) ℤ2 × ℤ300, which aligns with the decomposition obtained from its prime factorization according to the Fundamental Theorem of Finite Abelian Groups.
Step-by-step explanation:
The question asks to determine the isomorphism type of an Abelian group of order 600. To find this, we use the Fundamental Theorem of Finite Abelian Groups, which states that every finite Abelian group is isomorphic to a direct product of cyclic groups of prime power order. The integer 600 can be factored into prime powers as 600 = 23 × 31 × 52.
We then take each of the prime powers and express them as direct products of cyclic groups: 23 = 22 × 21, 31, and 52. Combining these gives us all possible cyclic decompositions for the group, and we can combine powers of the same prime together to get direct products like ℤ8 × ℤ3 × ℤ25, or any other product structure that results from partitioning the prime powers differently. For example, ℤ4 × ℤ2 × ℤ3 × ℤ25 is also an isomorphic representation. The correct answer thus matches the structure of one of these direct product decompositions.
Looking at the provided options, option (c) ℤ2 × ℤ300 presents a valid isomorphic structure because 300 can be further decomposed into ℤ4 × ℤ3 × ℤ25, which aligns with the decomposition obtained from the prime factorization.