The number of finite abelian groups of order n

 In the last post, we learned about how to classify finite abelian groups in two ways, and how much the classification systems rely on the prime factorization of n, the order of the group. Let me present a corollary to the classification standards.

Let the prime factorization of n be written as (p1^e1)(p2^e2)...(pk^ek) where all the values of e are positive integers. The number of different abelian groups of order n is the product of the exponents e1e2...ek.

 

For example, 30 = 2¹*3¹*5¹, also known as a square free number. There is only one finite abelian group of order 30. On the other hand, we can write 60 = 2²*3¹*5¹, so there are two different abelian groups of order 60, Z60 and Z2 ✕ Z30.

 

The fields of group theory and number theory are closely intertwined.