Counting Abelian Groups of Specific Order

How many Abelian groups are there of order $2^5 \times 3^6 \times 5^{20}$?

The problem of counting the number of Abelian groups of a given order delves into several important concepts of group theory. At the core of this problem is the classification of finite Abelian groups, which tells us that every finite Abelian group can be expressed as a direct sum of cyclic groups of prime power order. This concept is summarized succinctly in the Fundamental Theorem of Finite Abelian Groups, which provides a robust framework for dissecting the structure of such groups into simpler, well-understood components.

Solving this problem involves determining the number of distinct Abelian groups that share the given order, which is factored into powers of primes. It requires identifying all possible partitions of these prime powers as direct sums of cyclic components. For instance, given the prime factors 2 raised to the 5th power, 3 raised to the 6th power, and 5 raised to the 20th power, the task is to partition each into sums of integers representing the orders of cyclic subgroups. Each unique partition corresponds to a distinct group structure.

This exploration not only reinforces understanding of group structure and partitioning, but also illustrates the utility of the theory in classifying groups. It is an exercise in both combinatorial reasoning and algebraic insight, offering rich opportunities to explore the elegant interplay between number theory and abstract algebra.