Finite Abelian Groups

Products of prime-power cyclics; count = partitions.

The idea

The fundamental theorem of finite abelian groups describes every finite abelian group in terms of a single kind of building block.

Theorem (Fundamental theorem of finite abelian groups).

Every finite abelian group is isomorphic to a direct product of cyclic groups of prime-power order, $\mathbb{Z}/p_1^{k_1} \times \mathbb{Z}/p_2^{k_2} \times \cdots \times \mathbb{Z}/p_r^{k_r},$ and the list of prime powers $p_1^{k_1}, \dots, p_r^{k_r}$ is determined by the group up to reordering.

The theorem is a complete classification. Classifying all groups of a given order is out of reach in general, but among abelian groups nothing remains to find: cyclic groups of prime-power order are the building blocks, direct products are the only way they combine, and the uniqueness clause says two such products are isomorphic exactly when their lists of prime powers agree.

Because the answer is a list of prime powers with a prescribed product, classification becomes counting. An abelian group of order $p^{k}$ is a product $\mathbb{Z}/p^{k_1} \times \cdots \times \mathbb{Z}/p^{k_r}$ whose exponents are positive and sum to $k$, so the groups of order $p^{k}$ correspond exactly to the ways of writing $k$ as a sum of positive integers, with the order of the summands ignored. Counting abelian groups of a given order reduces to counting these sums. The figure lists the case $k = 4$: each way of writing $4$ as such a sum appears as a row of blocks, beside the group of order $p^{4}$ it names.

Ways to work on it

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