The six abelian groups of order 360 in both classification forms

Given: The objects and hypotheses in the example.

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).

For every finite abelian group $G$ there is a unique list $1<n_1\mid \cdots \mid n_r$ such that $G\cong C_{n_1}\times \cdots \times C_{n_r}$. Moreover $\mathrm{\mid }G\mathrm{\mid }=n_1\cdots n_r$. The trivial group corresponds to the empty list and empty product. (Fundamental theorem of finite abelian groups: invariant-factor form).

Let $n\ge 1$, and write its canonical prime factorisation as $n={\prod }_{i<r}{p}_i^{a_i}$, with the $p_i$ distinct and $a_i>0$. Then the number of isomorphism classes of abelian groups of order $n$ is ${\prod }_{i<r}P(a_i),$ where $P(a)$ is the number of partitions of $a$. For $n=1$ one has $r=0$, so the empty product is $1$. (The number of finite abelian groups of order n is the product of the partition numbers of the prime exponents of n).

Powers are the natural powers of def-group-power and finite products those of def-monoid-finite-product, both taken in the commutative monoid $(\mathbb{Z},\cdot ,1)$ of lem-units-of-z. Call $p:r\to \mathbb{Z}$ an injective list of primes when every $p_i$ is prime (def-prime) and $p_i=p_j$ forces $i=j$ (def-injection-surjection-bijection). Let $n\in \mathbb{Z}$ with $n\ge 1$ and let $p:r\to \mathbb{Z}$ be an injective list of primes such that every prime divisor of $n$ equals $p_i$ for some $i<r$. Then, with $v_q$ as in def-p-adic-valuation: 1. ${\displaystyle n=\prod _{i<r}{p}_i^{v_{p_i}(n)}}$; 2. $v_q(n)=0$ for every prime $q$ that is not among $p_0,\dots ,p_{r-1}$; 3. the exponents are determined by $n$: if $e:r\to \mathbb{N}$ and $n={\prod }_{i<r}{p}_i^{e_i}$, then $e_j=v_{p_j}(n)$ for every $j<r$. Clause 3 needs only injectivity of the list, not the covering hypothesis. (For $n\ge 1$ and any injective list $p:r\to \mathbb{Z}$ of primes containing every prime divisor of $n$, one has $n={\prod }_{i<r}{p}_i^{v_{p_i}(n)}$; the exponents are determined by $n$, and $v_q(n)=0$ for every prime $q$ outside the list).