Enumerate Isomorphic Groups of Order 16

Given a finite Abelian group G of order 16, enumerate all possible isomorphic groups by considering the factorization of order 16.

When dealing with the classification of finite Abelian groups, a fundamental concept rooted in group theory is that every finite Abelian group can be expressed as a direct sum of cyclic subgroups whose orders are powers of prime numbers. The problem of enumerating all isomorphic groups of order 16 is a specific instance where the order of the group is a perfect power of a prime number, specifically 2 to the power of 4. This allows us to approach the problem using the classification theorem of finitely generated Abelian groups.

The classification theorem asserts that a finite Abelian group of order n can be decomposed into a product of cyclic groups whose orders are powers of primes that divide n. In this case, since 16 is 2 raised to the power of 4, we can consider partitions of 4 to enumerate possible product forms. Specifically, we are looking at partitions that use powers of 2, such as (4), (3,1), (2,2), (2,1,1), and (1,1,1,1), which correspond to the group structures C16, C8xC2, C4xC4, C4xC2xC2, and C2xC2xC2xC2, respectively. These are the distinct isomorphic types of Abelian groups of order 16.

Understanding the decomposition of groups in terms of cyclic subgroups not only aids in classification but also in identifying characteristics such as the Sylow subgroups of the group, understanding automorphisms, and determining invariants used in group theory. This approach to solving the problem is pivotal for anyone studying algebraic structures, as it presents a powerful method of classifying and characterizing groups, leveraging partitions and fundamental theorems.