The additive rationals do not decompose as a product of finite cyclic prime-power groups

Given: The objects and hypotheses in the statement refuted.

On the set of pairs $(a,b)$ with $a,b\in \mathbb{Z}$ and $b\mathrm{\ne }0$, define $(a,b)\sim (c,d)⟺ad=cb\text{in }\mathbb{Z}\mathrm{.}$ This is an equivalence relation (lem-rat-equivalence). The rationals are the quotient $\mathbb{Q}$, and $[(a,b)]$ is written $a\mathrm{/}b$. (The rationals as equivalence classes of pairs of integers).

Natural exponents, in a monoid. Let $(M,\cdot ,e)$ be a monoid (def-semigroup-and-monoid) and $g\in M$. By the recursion theorem (thm-recursion), applied with the set $M$, the element $e$ and the function $x\mapsto x\cdot g$ from $M$ to $M$, there is exactly one function $\mathbb{N}\to M$, written $n\mapsto g^n$, with $g^0=e,g^{\sigma (n)}=g^n\cdot g(n\in \mathbb{N})\mathrm{.}$ In particular $g^0=e$ for every $g$, including $g=e$, and $g^1=g^{\sigma (0)}=e\cdot g=g$. Since $\mathbb{N}$ contains $0$ (def-natural-numbers), the exponent $0$ is a genuine value of the definition and not a separate convention. Integer exponents, in a group. Let $G$ be a group (def-group) and $g\in G$. Write $\iota :\mathbb{N}\to \mathbb{Z}$ for the embedding $k=[(k,0)]$ of lem-nat-embeds-int, which is injective, preserves addition, multiplication and order, and has as image exactly the nonnegative integers. For $x\in \mathbb{Z}$ define - $g^x:=g^k$, the natural power, when $0\le x$ and $x=k$; - $g^x:=(g^k{)}^{-1}$ when $x<0$ and $-x=k$. Why this is well defined. The order on $\mathbb{Z}$ is total and antisymmetric (thm-int-ordered-ring, def-int-order), so exactly one of $0\le x$ and $x<0$ holds and the two clauses never both apply. In the first clause $x$ is nonnegative, so $x=k$ for some $k\in \mathbb{N}$, and $k$ is unique because $\iota$ is injective. In the second clause $x<0$ gives $0=x+(-x)<0+(-x)=-x$ by compatibility of the order with addition (thm-int-ordered-ring, def-int-operations), so $-x$ is a positive integer and again $-x=k$ for a unique $k$. The inverse $(g^k{)}^{-1}$ is a single determined element by lem-inverse-unique and def-invertible-element. Finally the two readings of $g^k$, as a natural power and as an integer power, agree by construction, so no ambiguity is introduced. Abbreviation. In an exponent we write $k$ for the integer $k$ when a natural number $k$ is used where an integer is expected; this is unambiguous because $\iota$ is injective and preserves the arithmetic and the order, and because the two readings of $g^k$ agree as just noted. Additive notation. When the group is written additively the same object is written $ng$ or $n\cdot g$ rather than $g^n$, with $0g=0$ and $\sigma (n)g=ng+g$; the definitions are identical, only the symbols differ. (Powers $g^n$: natural exponents in a monoid and integer exponents in a group, with $g^0=e$).

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).