How many Abelian groups (up to isomorphism) are there a. of...

How many Abelian groups (up to isomorphism) are there
a. of order $6 ?$
b. of order $15 ?$
c. of order $42 ?$
d. of order $p q$, where $p$ and $q$ are distinct primes?
e. of order $p q r$, where $p, q$, and $r$ are distinct primes?
f. Generalize parts $\mathrm{d}$ and e.

Key Concepts

Fundamental Theorem of Finite Abelian Groups

This theorem states that every finite abelian group can be expressed as a direct sum of cyclic groups whose orders are powers of primes. It is the basis for classifying abelian groups up to isomorphism by breaking them down into simpler, well-understood building blocks, and it underlies the method used to count the number of abelian groups of a given order.

Prime Factorization of Group Order

Prime factorization is the process of expressing a number as a product of prime numbers. In the context of classifying finite abelian groups, the order of the group is factorized into primes, which then allows one to consider the group as a direct sum of its Sylow subgroups, each corresponding to one of these primes.

Primary Decomposition

Primary decomposition refers to the splitting of the abelian group into a direct sum of its primary components (subgroups corresponding to a given prime power). This step is crucial because the structure of the group in each primary component can be classified independently using partitions of the exponent, thereby simplifying the overall classification of the group.

Invariant Factor Decomposition

Invariant factor decomposition is a method to express a finite abelian group as a direct sum of cyclic groups where each cyclic component's order divides the order of the next one. This provides a unique classification scheme up to isomorphism and is often used to enumerate the distinct types of abelian groups of a specified order.

Direct Product of Cyclic Groups

The concept of a direct product of cyclic groups is central to the classification of finite abelian groups. It allows the group to be written as a product of simpler cyclic groups, and the number of nonisomorphic groups corresponds to the different ways the order’s prime power factors can be partitioned into cyclic factors.

Groups of Orders with Distinct Primes

When the order of an abelian group is the product of distinct primes, its primary components are all cyclic of prime order. This significantly simplifies the classification since, by the Chinese Remainder Theorem, the group is isomorphic to the direct product of its prime order components, often leading to a unique abelian group up to isomorphism in many cases.

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