How Many Abelian Groups of Order 81=3^4 Are There?
If p is a prime number, the Fundamental Theorem of Finite Abelian groups allows us to classify, up to isomorphism, the number of Abelian groups of order p^4. If p=3, then p^4=81. How many Abelian groups of order 81=3^4 are there? It's based on the number of integer partitions of the positive integer 4. There are 5 such partitions: 4, 3+1, 2+1+1, 1+1+1+1, and 2+2. The isomorphism classes of Abelian groups of order 81 are: 1) ℤ81, 2) ℤ3 ⊕ ℤ27, 3) ℤ3 ⊕ ℤ3 ⊕ ℤ9, 4) ℤ3 ⊕ ℤ3 ⊕ ℤ3 ⊕ ℤ3, and 5) ℤ9 ⊕ ℤ9.
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