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If $N$ is invertible in the base field, multiplication by $N$ on an abelian variety is finite étale: its differential is $N$ times the identity, and properness turns its zero-dimensional fibres into finite fibres. The Theorem of the Cube gives $[N{]}^{\ast }L\cong L^{N^2}$ for a symmetric ample line bundle. Top intersection products give degree $N^{2\dim A}$. For each prime power ${\ell }^r\mid N$, the kernels killed by ${\ell }^s$ have orders ${\ell }^{2s\dim A}$; the classification of finite abelian groups forces $2\dim A$ cyclic factors of length $r$. Combining the prime powers gives the displayed group.

A subset $A$ of a finite abelian group $G$ with subset density $\eta =|A|/|G|$ is quadratically $\alpha$-uniform, in the norm convention, if its balanced subset indicator $g=1_A-\eta$ has Gowers U3 norm at most $\alpha$. This controls correlations on additive three-dimensional cubes after the subset density contribution is removed. If a convention instead bounds the eighth power of the norm by $\alpha$, its parameter corresponds to the norm parameter ${\alpha }^{1/8}$. The convention must therefore be stated when giving quantitative thresholds.

For a finite abelian group $G$ with character group $\hat{G}$, the quantum Fourier transform isup to complex conjugation according to the transform convention.

For an abelian group $G$ and an integer $m\ge 1$, its $m$-torsion subgroup is $G[m]=\{x\in G:mx=0\}$, the kernel of multiplication by $m$. In a finite abelian group, every nonempty fiber of this homomorphism has cardinality $|G[m]|$, so $|G:mG|=|G[m]|$. This follows directly: if $mx=my$, then $x-y\in G[m]$, and conversely translating by an element of $G[m]$ leaves the image unchanged. On the cyclic group ${\mathbb{Z}}_N$, the solutions of $mx=0$ number $\gcd (m,N)$; divide both integers by their greatest common divisor to see that $x$ must be a multiple of $N/\gcd (m,N)$.

For an elliptic curve over a finite field of characteristic $p$, its rational-point group is finite and abelian. Prime-to-characteristic geometric torsion bounds every other prime-torsion dimension by two. Multiplication by $p$ has degree $p^2$ and inseparable degree at least $p$, so the $p$-torsion dimension is at most one. The structure theorem for finite abelian groups therefore gives two cyclic factors with $m\mid n$ and $p\nmid m$.