CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-11

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Isomorphism classes of abelian groups of order p^n are counted by partitions of n

Statement

For a prime $p$ and $n > 0$ , isomorphism classes of abelian groups of order $p^{n}$ are in bijection with partitions of $n$ . For $n = 0$ , the unique group is the trivial group and corresponds separately to the empty partition.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[L1]

Every finite abelian group is isomorphic to a finite direct product of cyclic groups of prime-power order. The multiset of their orders is uniquely determined by the group, up to permutation of the factors. (Fundamental theorem of finite abelian groups: elementary-divisor form).

[L2]

For $n > 0$ , a partition of $n$ is a finite nondecreasing list of positive integers $(e_{1}, \dots, e_{r})$ with $e_{1} + \dots + e_{r} = n$ , using finite natural sums as in def-nat-finite-sum-and-product and naturals as in def-natural-numbers. Equality is equality of these lists. The nondecreasing convention removes permutations from the data. (Partitions of a positive integer).

[L3]

If $G$ and $H$ are finite groups, then their external direct product is finite and has order $∣ G \times H ∣ = ∣ G ∣ ∣ H ∣$ . (For finite groups $G$ and $H$ , $∣ G \times H ∣ = ∣ G ∣ ∣ H ∣$ ).

Proof

technique · direct

1.1

The elementary-divisor theorem writes such a group uniquely as $C_{p^{e_{1}}} \times \dots \times C_{p^{e_{r}}}$ with the positive exponents arranged nondecreasingly. The product-order formula gives $e_{1} + \dots + e_{r} = n$ .

givenL1L2L3

2.1

Thus the exponents form a partition of $n$ , and every partition constructs a group of order $p^{n}$ . Uniqueness of elementary divisors makes the two constructions inverse.

step 1.1∎

Depends on

Used by

Dependency tree · two levels

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Sources

Keith Conrad, Decomposition of Finite Abelian Groups, §§1-4 (standard reference, not scraped)
Richard Elman, Lectures on Abstract Algebra, Ch. 14 (standard reference, not scraped)