For every natural $n$, $(\mathbb{Z}\mathrm{/}n,+)$ is an abelian group, multiplication is a commutative monoid operation, and both distributive laws hold

Depends on

• Addition and multiplication on $\mathbb{Z}/n$ by $[a]_n+[b]_n=[a+b]_n$ and $[a]_n[b]_n=[ab]_n$
• The integers form a commutative ring
• Group and abelian group
• Semigroup and monoid

Used by

• The discriminant counts roots of Ax²+Bx+C≡0 (mod p) for odd prime p∤ A Corollary
• The Galois group of a cyclotomic extension is abelian Corollary
• In ℤ/4 the sets A=B={0,2} have | A+B|=2, below the Cauchy–Davenport bound 3 Counterexample
• Over ℤ/2, an antisymmetric bilinear form need not be alternating Counterexample
• The action of ℤ/2 on two disjoint two-point orbits is free but not transitive Counterexample
• ℤ/1 has one element and satisfies [0]₁=[1]₁, so it is not a field Counterexample
• The commutator pairing of an extraspecial p-group relative to a chosen generator of its centre Definition
• The Heisenberg group of order p³ over ℤ/p Definition
• The unit group (ℤ/n)^× and Euler's totient φ(n)=|(ℤ/n)^×| for n≥1 Definition
• ⟨ a,b∣ a², b², aba⁻¹b⁻¹⟩≅(ℤ/2)×(ℤ/2) Example
• ⟨ a∣ aⁿ⟩≅(ℤ/n,+) for every n≥ 1 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
• Assuming the Axiom of Choice, the Chinese-remainder map Z/6Z -> Z/2Z direct-sum Z/3Z is an isomorphism by local tests Example
• Dₙ≅⟨ r,s∣ rⁿ, s², srs⁻¹r⟩ for the dihedral group Dₙ=⟨{ρ,σ}⟩leqSym(ℤ/n), n≥ 3 Example
• Hom_ℤ(ℤ/m,ℤ/n)≅ℤ/gcd(m,n) for n≥1 Example
• Nonzero constant series can multiply to zero in (ℤ/4ℤ)⟦ x⟧ Example
• Over Z, the ideal (2) acts surjectively on Z/3Z but does not kill it Example
• Row operations track determinant correctly for a singular triangular matrix over ℤ/6 Example
• The action of ℤ/6 on the cosets of {0,3} is transitive with kernel {0,3} and is not faithful Example
• The four rotations of a square act freely, transitively and faithfully on its vertices Example
• The Klein four-group as the direct product of two groups of order 2 Example
• The residue classes ℤ/n as a ℤ-module Example
• The ring (ℤ/2)^ℕ is not Noetherian Example
• The trivial action of ℤ/2 on a singleton is transitive but not faithful Example
• There are six binary necklaces of length four up to rotation Example
• Two filtered abelian groups with the same associated graded 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
• Abelian-group model for spectral-sequence computations Lemma
• An elementary abelian p-group has a canonical Fₚ-vector-space structure Lemma
• Cyclic shifting is an action of ℤ/m on the words of length m over a set Lemma
• Every finite abelian group is a quotient of (ℤ/n)ᵏ for some n and k Lemma
• If gcd(‖ a‖,m)=1 then the shift stabiliser of a is trivial, so its orbit has exactly m elements Lemma
• Raising to the power 1+p is an automorphism of order p of a cyclic group of order p² Lemma
• There are arbitrarily large primes congruent to 2 modulo 3 Lemma
• Every nonzero residue modulo an odd prime is a sum of two squares Proposition
• For every n∈ℕ, the congruence-class group (ℤ/n,+) is the quotient group (ℤ,+)/nℤ Proposition
• For every n∈ℕ, the congruence-class ring ℤ/n is the quotient ring ℤ/nℤ Proposition

…and 9 more results.

Dependency tree · two levels

22 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)