ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passverified 2026-08-03 (gpt-5.6-sol-codex-subscription)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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For $n \geq 1$ , the cosets of $n Z$ are the $n$ congruence classes modulo $n$

Example

Let $n \geq 1$ be an integer. In the additive group $(Z, +)$ , the subgroup $n Z$ has left cosets

$a + n Z = {x \in Z : x \equiv a (m o d n)} .$

There are exactly $n$ such cosets, represented uniquely by $0, 1, \dots, n - 1$ , and $[Z : n Z] = n$ .

Facts & Assumptions

Given: A positive integer $n$ and the additive group $Z$ .

[L1]

The set $n Z = {n k : k \in Z}$ is a subgroup of $(Z, +)$ (Every subgroup of $(Z, +)$ is $⟨ n ⟩ = n Z$ for exactly one natural number $n$ , The integers form a commutative ring).

[F1]

In additive notation, the coset represented by $a$ is $a + n Z$ (Left and right cosets $g H$ and $H g$ of a subgroup).

[F2]

The congruence $x \equiv a (m o d n)$ means $n ∣ (x - a)$ (Congruence modulo an integer: $a \equiv b (m o d n)$ when $n ∣ (a - b)$ , including the moduli $0$ and $1$ ).

[L2]

Every class modulo positive $n$ has exactly one representative $r$ with $0 \leq r < n$ , and there are $n$ classes (For $n \geq 1$ , every class in $Z / n$ has one representative $r$ with $0 \leq r < n$ , so $∣ Z / n ∣ = n$ ; while $Z / 0$ is in bijection with $Z$ ).

[F3]

The index is the finite cardinality of the coset set (The coset set $G / H$ and the index $[G : H]$ of a subgroup, The left cosets of a subgroup partition the group).

Verification

technique · direct

1.1

By [L1], $n Z$ is a subgroup of $(Z, +)$ . For $x \in Z$ , one has $x \in a + n Z$ exactly when $x = a + n k$ for some integer $k$ , equivalently $n ∣ (x - a)$ , equivalently $x \equiv a (m o d n)$ .

L1F1F2

2.1

By [L2], every such class has a unique representative in ${0, \dots, n - 1}$ , and distinct representatives give distinct cosets by step 1.1.

step 1.1L2

3.1

Thus the coset set has cardinality $n$ , so $[Z : n Z] = n$ .

step 2.1F3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

41 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

UCL lecture notes, Cosets and Lagrange's theorem (standard reference, not scraped)
Thomas W. Judson, Abstract Algebra: Theory and Applications, §6.1: Cosets (standard reference, not scraped)

For n≥1n≥1, the cosets of nZnZ are the nn congruence classes modulo nn