Classification of Groups of Order 540

Using the Fundamental Theorem of Finite Abelian Groups, list all the groups of order 540 by considering its prime factorization.

The problem of classifying groups of a given order often involves deep insights into group structure, particularly when using the Fundamental Theorem of Finite Abelian Groups. This theorem tells us that any finite abelian group can be expressed as a direct product of cyclic groups of prime power order. When approaching a problem like this, one of the first tasks is to determine the prime factorization of the order, which in this case is 540. The prime factorization serves as a guide to understanding the possible structures of the groups involved. Here, 540 can be decomposed into the prime factors 2, 3, and 5, specifically $2^2$, $3^3$, and 5. Each prime factor corresponds to a cyclic group and the exponent indicates the number of times that cyclic group needs to divide the total order. The crucial process in this problem involves constructing all possible products of cyclic groups that result in the given group order, essentially leveraging the different combinations of the prime power components. Recognizing the interplay between these cyclic subgroups and how they combine to form a larger group is key in group theory, revealing the intrinsic symmetry and structural patterns within the group. Students should focus on understanding how these cyclic compositions adhere to the rules of group theory and how the structure theorem can simplify seemingly complex organizational possibilities into more manageable forms.