CorollaryStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck 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 $(e_1,\ldots,e_r)$ with $e_1+\cdots+e_r=n$ , 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 $C_{p^{e_1}}\times\cdots\times C_{p^{e_r}}$ with the positive exponents arranged nondecreasingly. The product-order formula gives $e_1+\cdots+e_r=n$ .

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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 59 results over 21 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

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)