How many abelian groups are in order 144?

There are exactly 10 abelian groups of order 144.

Understanding Abelian Groups

An abelian group is a fundamental concept in abstract algebra, defined as a group in which the result of applying the group operation to two elements does not depend on the order in which they are written (i.e., the operation is commutative). These groups are named after the Norwegian mathematician Niels Henrik Abel.

Finite abelian groups possess a highly structured and well-understood classification, primarily governed by the Fundamental Theorem of Finite Abelian Groups. This theorem states that every finite abelian group can be uniquely expressed as a direct product of cyclic groups of prime-power order. This unique decomposition allows for a systematic way to count such groups for any given order.

Deriving the Number of Abelian Groups of Order 144

To determine the number of distinct abelian groups of a specific order $n$, we first find the prime factorization of $n$. For each prime power in the factorization, we count the number of ways its exponent can be partitioned. The total number of abelian groups is then the product of these partition counts.

Let's apply this to the order 144:

1. Prime Factorization: The first step is to find the prime factorization of 144.
   $144 = 12 \times 12 = (2^2 \times 3) \times (2^2 \times 3) = 2^4 \times 3^2$.

2. Partitions of Exponents:

   • For the prime factor 2 (exponent 4): We need to find the number of partitions of 4. A partition of an integer is a way of writing it as a sum of positive integers, where the order of the summands does not matter. The partitions of 4 are:

     • 4 (corresponding to the cyclic group $Z_{16}$)
     • 3 + 1 (corresponding to $Z_8 \times Z_2$)
     • 2 + 2 (corresponding to $Z_4 \times Z_4$)
     • 2 + 1 + 1 (corresponding to $Z_4 \times Z_2 \times Z_2$)
     • 1 + 1 + 1 + 1 (corresponding to $Z_2 \times Z_2 \times Z_2 \times Z_2$)
       There are 5 partitions of 4. This is denoted as $P(4) = 5$.

   • For the prime factor 3 (exponent 2): We need to find the number of partitions of 2.

     • 2 (corresponding to the cyclic group $Z_9$)
     • 1 + 1 (corresponding to $Z_3 \times Z_3$)
       There are 2 partitions of 2. This is denoted as $P(2) = 2$.

3. Total Number of Abelian Groups: The total number of non-isomorphic abelian groups of order 144 is the product of the number of partitions for each prime power:
   Total Abelian Groups = $P(4) \times P(2) = 5 \times 2 = 10$.

Therefore, there are exactly 10 distinct abelian groups of order 144. Each of these groups can be uniquely represented as a direct product of cyclic groups, such as $Z_{16} \times Z_9$ or $Z_4 \times Z_2 \times Z_2 \times Z_3 \times Z_3$.

Types of Groups of Order 144

While this question focuses specifically on abelian groups, it's insightful to understand how they fit within the broader classification of groups of order 144. Group theory classifies groups based on various properties like commutativity (abelian), nilpotency, solvability, and simplicity.

For groups of order 144, we observe the following quantities for different classifications:

This table illustrates that while there are 10 abelian groups, there are many more non-abelian groups of order 144, including those that are nilpotent and solvable. The fact that there are no simple groups of order 144 indicates that all groups of this order can be broken down into smaller groups through normal subgroups.