For $n\ge 1$, every class in $\mathbb{Z}\mathrm{/}n$ has one representative $r$ with $0\le r<n$, so $\mid \mathbb{Z}\mathrm{/}n\mid =n$; while $\mathbb{Z}\mathrm{/}0$ is in bijection with $\mathbb{Z}$

Depends on

• The congruence class $[a]_n$ and the quotient set $\mathbb{Z}/n$
• Division with remainder in $\mathbb{Z}$: for $a \in \mathbb{Z}$ and $b > 0$ there are unique $q, r \in \mathbb{Z}$ with $a = qb + r$ and $0 \le r < b$
• The cardinality $\lvert A\rvert$ of a finite set
• Injection, surjection, bijection
• The natural numbers $\mathbb{N}$ (von Neumann)
• The naturals embed in the integers

Used by

• Modular order finding with rational reconstruction Algorithm
• Shor factoring with verified output and bounded outer retries Algorithm
• Connected coverings of the circle are classified by the subgroups nℤ for n≥0 Corollary
• φ(1)=1, and φ(p)=p-1 for every prime p Corollary
• 2ℤ has index 2 in ℤ and is nevertheless equinumerous with ℤ Counterexample
• Cyclic convolution wraps a high coefficient without zero padding Counterexample
• The action of ℤ/2 on two disjoint two-point orbits is free but not transitive Counterexample
• The short exact sequence to Sₙ does not prove Bₙ torsion-free Counterexample
• Visible Artin relations alone do not prove presentation completeness Counterexample
• ℤ/1 has one element and satisfies [0]₁=[1]₁, so it is not a field Counterexample
• Denominator-bounded rational phase reconstruction Definition
• The counting inner product on ℂ^ℤ/Nℤ Definition
• The unit group (ℤ/n)^× and Euler's totient φ(n)=|(ℤ/n)^×| for n≥1 Definition
• The unitary discrete Fourier transform on ℤ/Nℤ Definition
• The unnormalised cyclic convolution on ℤ/Nℤ Definition
• (ℤ/12)^×={[1],[5],[7],[11]} and φ(12)=4 Example
• (ℤ/8)^×={[1],[3],[5],[7]} is not cyclic because every element squares to [1] Example
• ⟨ a,b∣ a², b², aba⁻¹b⁻¹⟩≅(ℤ/2)×(ℤ/2) Example
• ⟨ a∣ aⁿ⟩≅(ℤ/n,+) for every n≥ 1 Example
• A modulus n>1 is composite exactly when two nonzero classes in ℤ/n have product [0]ₙ Example
• A=B={0,1,2} in ℤ/7: the sumset has five elements and the bound is tight Example
• An involution on five points has three fixed points and one two-point orbit, verifying 5≡3 (mod 2) Example
• Cyclic convolution on ℤ/4ℤ via the DFT Example
• Dₙ≅⟨ r,s∣ rⁿ, s², srs⁻¹r⟩ for the dihedral group Dₙ=⟨{ρ,σ}⟩leqSym(ℤ/n), n≥ 3 Example
• For n≥1, the cosets of nℤ are the n congruence classes modulo n Example
• The 3×3 upper-unitriangular group over a prime field has generator rank two Example
• The action of ℤ/6 on the cosets of {0,3} is transitive with kernel {0,3} and is not faithful Example
• The Chinese remainder bijection ℤ/12→ℤ/3×ℤ/4 computed on all standard representatives Example
• The finite Heisenberg group is the unique Sylow p-subgroup of its coordinate upper-triangular group Example
• The five characters of Z/5Z and their orthogonality Example
• The four cosets of 4ℤ in (ℤ,+) reproduce addition modulo 4 Example
• The four rotations of a square act freely, transitively and faithfully on its vertices Example
• The Frattini subgroup of (ℤ/p)ⁿ is trivial Example
• The two-circle wedge has both regular and nonregular connected three-sheeted coverings Example
• There are six binary necklaces of length four up to rotation Example
• False: m⊗ n=0 implies m=0 or n=0 False statement
• False: tensoring preserves injections False statement
• False: ℤ/m⊗_ℤℤ/n is nonzero for all positive m,n False statement
• A cyclic group of order n has exactly φ(n) generators Lemma
• A polynomial-size prime field meets the soundness budget Lemma

…and 29 more results.

Dependency tree · two levels

33 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• K. Conrad, Modular Arithmetic (standard reference, not scraped)